Introduction to Statistics Using OpenOffice.org Calc.en

Introduction to Statistics Using
OpenOffice.org Calc
Dana Lee Ling
College of Micronesia – FSM
Pohnpei, Federated States of Micronesia
2008 College of Micronesia-FSM. This work is licensed under a Creative Commons Attribution 3.0
License. This work is licensed under the Creative Commons Attribution 3.0 License. To view a copy of this license,
visit http://creativecommons.org/licenses/by/3.0/ or send a letter to Creative Commons, 171 Second Street, Suite 300,
San Francisco, California, 94105, USA.
Printed in the Federated States of Micronesia
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Contents
Introduction: Samples and Levels of Measurement........................................................................................................5
1.1 Populations and Samples..........................................................................................................................................5
1.2 Simple random samples............................................................................................................................................6
1.3 Experimental Design.................................................................................................................................................8
Visualizing data.................................................................................................................................................................10
2.1 Graphs and Charts...................................................................................................................................................10
2.2 Histograms and Frequency Distributions................................................................................................................13
2.3 Shapes of Distributions...........................................................................................................................................19
Endnote: Creating histograms with spreadsheets..........................................................................................................20
Measures of Middle and Spread......................................................................................................................................28
3.1 Measures of central tendency: mode, median, mean, midrange.............................................................................28
3.2 Differences in the Distribution of Data...................................................................................................................29
3.3 Variables..................................................................................................................................................................31
3.4 A Measure of Relative Standing..............................................................................................................................32
Paired Data and Scatter Diagrams..................................................................................................................................34
4.1 Best Fit Lines: Linear Regressions.........................................................................................................................34
4.2 Slope and Intercept..................................................................................................................................................35
4.25 Creating an xy scattergraph using OpenOffice.org Calc 2.4, 3.0..........................................................................35
4.3 Relationships between variables.............................................................................................................................40
4.4 Correlation..............................................................................................................................................................41
Introduction to Probability..............................................................................................................................................46
5.1 Ways to determine a probability.............................................................................................................................46
5.2 Sample space...........................................................................................................................................................47
5.3 Relative Frequency.................................................................................................................................................48
5.4 Or............................................................................................................................................................................50
Probability Distributions..................................................................................................................................................53
6.1 Types of probabilities and distributions..................................................................................................................53
6.2 Calculations of the mean and the standard deviation..............................................................................................54
Introduction to the Normal Distribution........................................................................................................................57
7.1 Distribution shape...................................................................................................................................................57
7.2 Seven Pennies.........................................................................................................................................................57
7.3 The Normal Curve..................................................................................................................................................59
7.4 The area to the left of x...........................................................................................................................................64
7.5 Area to x..................................................................................................................................................................66
Sampling Distribution of the Mean.................................................................................................................................68
8.1 Distribution of Statistics..........................................................................................................................................68
8.2 Central Limit Theorem............................................................................................................................................71
Confidence Intervals.........................................................................................................................................................73
9.1 Inference and Point Estimates.................................................................................................................................73
9.12 Introduction to Confidence Intervals....................................................................................................................74
9.18 Confidence Intervals for n > 30 where σ is known...............................................................................................75
Confidence intervals for n ≥ 5 using sx........................................................................................................................76
9.3 Confidence intervals for a proportion.....................................................................................................................78
9.4 Deciding on a sample size.......................................................................................................................................79
Hypothesis Testing............................................................................................................................................................81
10.1 Confidence Interval Testing..................................................................................................................................81
10.2 Hypothesis Testing................................................................................................................................................83
10.3 P-value..................................................................................................................................................................86
10.4 One Tailed Tests....................................................................................................................................................88
10.5 Hypothesis test for a proportion............................................................................................................................89
Testing the two sample means with the t-test.................................................................................................................92
11.1 Paired differences: Dependent samples.................................................................................................................92
11.2 T-test for means for independent samples.............................................................................................................94
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Technology note
This statistics text utilizes OpenOffice.org Calc to make statistical calculations. This text also includes references to
Microsoft Excel functions where they differ from OpenOffice.org Calc functions. Screen shots, however, are almost
always from OpenOffice.org Calc. Gnumeric spreadsheet users can also use this text following the Microsoft Excel
function formats.
The text does not use any add-ins, add-ons, statistical extensions, or separate dedicated statistical packages. This choice
is very deliberate. Course alumni are most likely to encounter default installations of spreadsheet software without such
additional software. The introduction to statistics alumni should not feel that they cannot "do" statistics because they
lack a special add-in or dedicated package function that may require administrative privileges to install if even
available. Given an "out-of-the-box" installation of a spreadsheet, alumni should be able to generate and use statistics.
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Introduction: Samples and Levels of Measurement
1.1 Populations and Samples
Statistics studies groups of people, objects, or data measurements and produces summarizing mathematical information
on the groups. The groups are usually not all of the possible people, objects, or data measurements. The groups are
called samples. The larger collection of people, objects or data measurements is called the population.
Statistics attempts to predict measurements for a population from measurements made on the smaller sample. For
example, to determine the average weight of a student at the college, a study might select a random sample of fifty
students to weigh. Then the measured average weight could be used to estimate the average weight for all student at the
college. The fifty students would be the sample, all students at the college would be the population.
Population: The complete group of elements, objects, observations, or people.
• Parameters: Measurements of the population.
Sample: A part of the population. A sample is usually more than five measurements, observations, objects, or people,
and smaller than the complete population.
• Statistics: Measurements of a sample.
Examples
We could use the ratio of females to males in a class to estimate the ratio of females to males on campus. The sample is
the class. The population is all students on campus.
We could use the average body fat index for a randomly selected group of females between the ages of 18 and 22 on
campus to determine the average body fat index for females in the FSM between the ages of 18 and 22. The sample is
those females on campus that we've measured. The population is all females between the ages of 18 and 22 in the FSM.
Measurements are made of individual elements in a sample or population. The elements could be objects, animals, or
people.
Sample size n
The sample size is the number of elements or measurements in a sample. The lower case letter n is used for sample size.
If the population size is being reported, then an upper case N is used. The spreadsheet function for calculating the
sample size is the COUNT function.
=COUNT(data)
Types of measurement
Qualitative data refers to descriptive measurements, typically non-numerical.
Quantitative data refers to numerical measurements. Quantitative data can be discrete or continuous.
•
•
Discrete: A countable or limited number of possible numeric values.
Continuous: An infinite number of possible numeric values.
Levels of measurement
Type
Subtype
Qualitative
Q
u
a
n
t
i
t
a
Discrete
Level of
measurement
Nominal
Ordinal
Continuous Interval
Definition
Examples
Sorting by categories such as red, orange, yellow,
green, blue, indigo, violet
Grading systems: A, B, C, D, F
In rank order, there
Sakau market rating system where the number of cups
exists an order but
until one is pwopihda...
differences and ratios
(highest),
,
,
,...
have no meaning
(lowest)
Differences have
The numbering of the years: 2001, 2000, 1999. The
meaning, but not ratios. year 2000 is 1000 years after 1000 A.D. (the difference
There is either no zero has meaning), but it is NOT twice as many years (the
In name only
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Type
Subtype
Level of
measurement
Definition
Examples
ratio has no meaning). Someone born in 1998 is eight
years younger than someone born in 1990: 1998 −
1990. A vase made in 2000 B.C., however, is not twice
or the zero has no
as old as a vase made in 1000 B.C. The complication is
mathematical meaning. subtle and basically can stem from two sources: either
there is no zero or the zero is not a true zero. The
Fahrenheit and Celsius temperature systems both suffer
t
from the later defect.
i
v
Difference and ratios
Physical quantities: distance, height, speed, velocity,
e
have meaning. There is
Ratio
time in seconds, altitude, acceleration, mass,... 100 kg
a mathematically
is twice as heavy as 50 kg. Ten dollars is 1/10 of $100.
meaningful zero
Descriptive statistics: Numerical or graphical representations of samples or populations. Can include numerical
measures such as mode, median, mean, standard deviation. Also includes images such as graphs, charts, visual linear
regressions.
Inferential statistics: Using descriptive statistics of a sample to predict the parameters or distribution of values for a
population.
1.2 Simple random samples
The number of measurements, elements, objects, or people in a sample is the sample size n. A simple random sample of
n measurements from a population is one selected in a way that:
•
•
any member of the population is equally likely to be selected.
any sample of a given size is equally likely to be selected.
Ensuring that a sample is random is difficult. Suppose I want to study how many Pohnpeians own cars. Would people I
meet/poll on main street Kolonia be a random sample? Why? Why not?
Studies often use random numbers to help randomly selects objects or subjects for a statistical study. Obtaining random
numbers can be more difficult than one might at first presume.
Computers can generate pseudo-random numbers. "Pseudo" means seemingly random but not truly random. Computer
generated random numbers are very close to random but are actually not necessarily random. Next we will learn to
generate pseudo-random numbers using a computer. This section will also serve as an introduction to functions in
spreadsheets.
Coins and dice can be used to generate random numbers.
Using a spreadsheet to generate random numbers
This course presumes prior contact with a course such as CA 100 Computer Literacy where a basic introduction to
spreadsheets is made.
The random function RAND generates numbers between 0 and 0.9999...
=rand()
The random number function consists of a function name, RAND, followed by parentheses. For the random function
nothing goes between the parentheses, not even a space.
To get other numbers the random function can be multiplied by coefficient. To get whole numbers the integer function
INT can be used to discard the decimal portion.
=INT(argument)
The integer function takes an "argument." The argument is a computer term for an input to the function. Inputs could
include a number, a function, a cell address or a range of cell addresses. The following function when typed into a
spreadsheet that mimic the flipping of a coin. A 1 will be a head, a 0 will be a tail.
=INT(RAND()*2)
The spreadsheet can be made to display the word "head" or "tail" using the following code in OpenOffice.org Calc:
=CHOOSE(INT(RAND()*2);"head";"tail")
Note that in OpenOffice.org Calc the formula uses semi-colons.
A similar function will do the same in Microsoft Excel, the only difference is that Excel uses commas:
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=CHOOSE(INT(RAND()*2),"head","tail")
A single die can also be simulated using the following function
=INT(6*RAND()+1)
To randomly select among a set of student names, the following model can be built upon. Note the formula uses
OpenOffice.org Calc semi-colons.
=CHOOSE(INT(RAND()*6+1);"Jen";"John";"Jess";"Jeff";"Jocelyn";"Jim")
To generate another random choice, press the F9 key on the keyboard. F9 forces a spreadsheet to recalculate all
formulas.
Methods of sampling
When practical, feasible, and worth both the cost and effort, measurements are done on the whole population. In many
instances the population cannot be measured. Sampling refers to the ways in which random subgroups of a population
can be selected. Some of the ways are listed below.
Census: Measurements done on the whole population.
Sample: Measurements of a representative random sample of the population.
Simulation
Today this often refers to constructing a model of a system using mathematical equations and then using computers to
run the model, gathering statistics as the model runs.
Stratified sampling
To ensure a balanced sample: Suppose I want to do a study of the average body fat of young people in the FSM. The
FSM population is roughly half Chuukese, but the Palikir campus population is more than half Pohnpeian. If I choose as
my sample students at the Palikir campus, then I am likely to wind up with Pohnpeians being over represented relative
to the actual national proportion of Pohnpeians.
Fractional share of national Number of student seats held
Fractional share of the
population (relative
by state at the national
national campus student
frequency)
campus
seats
Chuuk
53595
0.501
679
0.20
Kosrae 7686
0.072
316
0.09
Pohnpei 34486
0.322
2122
0.62
Yap
11241
0.105
287
0.08
107008
1.00
3404
1.00
The solution is to use stratified sampling. First I decide I want 100 students that are representative of the four states.
Then I can randomly pick 50 Chuukese students, 7 Kosraen, 32 Pohnpeian, and 11 Yapese and I will accurately reflect
the makeup of the nation rather than the national campus. Each state would be considered a single strata.
State
Population
Systematic sampling
Used where a population is in some sequential order. A start point must be randomly chosen. Useful in a measuring a
timed event. Never used if there is a cyclic or repetitive nature to a system: If the sample rate is roughly equal to the
cycle rate, then the results are not going to be randomly distributed measurements. For example, suppose one is
studying whether the sidewalks on campus are crowded. If one measures during the time between class periods when
students are moving to their next class - then one would conclude the sidewalks are crowded. If one measured only
when classes were in session, then one would conclude that there is no sidewalk crowding problem. This type of
problem in measurement occurs whenever a system behaves in a regular, cyclical manner. The solution would be ensure
that the time interval between measurements is random.
Cluster sampling
The population is divided into naturally occurring subunits and then subunits are randomly selected for measurement. In
this method it is important that subunits (subgroups) are fairly interchangeable. Suppose we want to poll the people in
Kitti's opinion on whether they would pay for water if water was guaranteed to be clean and available 24 hours a day.
We could cluster by breaking up the population by kosapw and then randomly choose a few kosapws and poll everyone
in these kosapws. The results could probably be generalized to all Kitti.
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Convenience sampling
Results or data that are easily obtained is used. Highly unreliable as a method of getting a random samples. Examples
would include a survey of one's friends and family as a sample population. Or the surveys that some newspapers and
news programs produce where a reporter surveys people shopping in a store.
1.3 Experimental Design
In science, statistics are gathered by running an experiment and then repeating the experiment. The sample is the
experiments that are conducted. The population is the theoretically abstract concept of all possible runs of the
experiment for all time.
The method behind experimentation is called the scientific method. In the scientific method, one forms a hypothesis,
makes a prediction, formulates an experiment, and runs the experiment.
Some experiments involve new treatments, these require the use of a control group and an experimental group, with the
groups being chosen randomly and the experiment run double blind. Double blind means that neither the experimenter
nor the subjects know which treatment is the experimental treatment and which is the control treatment. A third party
keeps track of which is which usually using number codes. Then the results are tested for a statistically significant
difference between the two groups.
Placebo effect: just believing you will improve can cause improvement in a medical condition.
Replication is also important in the world of science. If an experiment cannot be repeated and produce the same results,
then the theory under test is rejected.
Some of the steps in an experiment are listed below:
1. Identify the population of interest
2. Specify the variables that will be measured. Consider protocols and procedures.
3. Decide on whether the population can be measured or if the measurements will have to be on a sample of the
population. If the later, determine a method that ensures a random sample that is of sufficient size and
representative of the population.
4. Collect the data (perform the experiment).
5. Analyze the data.
6. Write up the results and publish! Note directions for future research, note also any problems or complications
that arose in the study.
Observational study
Observational studies gather statistics by observing a system in operation, or by observing people, animals, or plants.
Data is recorded by the observer. Someone sitting and counting the number of birds that land or take-off from a bird
nesting islet on the reef is performing an observational study.
Surveys
Surveys are usually done by giving a questionnaire to a random sample. Voluntary responses tend to be negative. As a
result, there may be a bias towards negative findings. Hidden bias/unfair questions: Are you the only crazy person in
your family?
Generalizing
The process of extending from sample results to population. If a sample is a good random sample, representative of the
population, then some sample statistics can be used to estimate population parameters. Sample means and proportions
can often be used as point estimates of a population parameter.
Although the mode and median, covered in chapter three, do not always well predict the population mode and median,
there there situations in which a mode may be used. If a good, random, and representative sample of students finds that
the color blue is the favorite color for the sample, then blue is a best first estimate of the favorite color of the population
of students or any future student sample.
Favorite color
Blue
Black
White
Green
Frequency f
32
18
10
9
Favorite colors
Relative Frequency or p(color)
35%
20%
11%
10%
-8-
Favorite color Frequency f
Relative Frequency or p(color)
Red
6
7%
Pink
5
5%
Brown
4
4%
Gray
3
3%
Maroon
2
2%
Orange
1
1%
Yellow
1
1%
Sums:
91
100%
If the above sample of 91 students is a good random sample of the population of all students, then we could make a
point estimate that roughly 35% of the students in the population will prefer blue.
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Visualizing data
2.1 Graphs and Charts
The table below includes FSM census 2000 data and student seat numbers for the national site of COM-FSM circa
2004.
State
Chuuk
Kosrae
Pohnpei
Yap
Population
(20000
53595
7686
34486
11241
107008
Fractional share of national
population (relative
frequency)
0.5
0.07
0.32
0.11
1
Number of student seats
held by state at the national
campus
679
316
2122
287
3404
Fractional share of the
national campus student
seats
0.2
0.09
0.62
0.08
1
Circle or pie charts
In a circle chart the whole circle is 100% Used when data adds to a whole, e.g. state populations add to yield national
population.
A pie chart of the state populations:
Column charts
Column charts are also called bar graphs. A column chart of the student seats held by each state at the national site:
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Pareto chart
If a column chart is sorted so that the columns are in descending order, then it is called a Pareto chart. Descending order
means the largest value is on the left and the values decrease as one moves to the right. Pareto charts are useful ways to
convey rank order as well as numerical data.
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Line graph
A line graph is a chart which plots data as a line. The horizontal axis is usually set up with equal intervals. Line graphs
are not used in this course and should not be confused with xy scattergraphs.
XY Scatter graph
When you have two sets of continuous data (value versus value, no categories), use an xy graph. These will be covered
in more detail in the chapter on linear regressions.
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2.2 Histograms and Frequency Distributions
A distribution counts the number of elements of data in either a category or within a range of values. Plotting the count
of the elements in each category or range as a column chart generates a chart called a histogram. The histogram shows
the distribution of the data. The height of each column shows the frequency of an event. This distribution often provides
insight into the data that the data itself does not reveal. In the histogram below, the distribution for male body fat among
statistics students has two peaks. The two peaks suggest that there are two subgroups among the men in the statistics
course, one subgroup that is at a healthy level of body fat and a second subgroup at a higher level of body fat.
The ranges into which values are gathered are called bins, classes, or intervals.
Nominal level of measurement
At the nominal level of measurement one can determine the frequency of elements in a category, such as students by
state in a statistics course.
State
Chuuk
Kosrae
Pohnpei
Yap
Sums:
Frequency
6
6
31
11
54
Rel Freq
0.11
0.11
0.57
0.20
1,00
Ordinal level of measurement
Unbinned
At the ordinal level, a frequency distribution can be done using the rank order, counting the number of elements in each
rank order to obtain a frequency. When the frequency data is calculated in this way, the distribution is unbinned, not
aggregated.
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Age
17
18
19
20
21
22
23
24
25
26
27
sums
Frequency
1
5
14
12
9
1
3
3
1
1
1
51
Rel Freq
0.02
0.1
0.27
0.24
0.18
0.02
0.06
0.06
0.02
0.02
0.02
1
Binned
The ranks can be collected together, binned, to reduce the number of rank order categories. in the example below the
age data in gathered into two-year cohorts.
Age
19
21
23
25
27
Sums:
Frequency
20
21
4
4
2
51
Rel Freq
0.39
0.41
0.08
0.08
0.04
1
Ratio level of measurement
At the ratio level data is always gathered into ranges. At the ratio level, binned histograms are used. Ratio level data is
not necessarily in a finite number of ranks as was ordinal data.
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The ranges into which data is gathered are defined by a bin lower limit and a bin upper limit. The width is the bin upper
limit minus the bin lower limit. In this class histograms are generated using the bin upper limit in conformance with the
usage in spreadsheets.
To calculate the bin lower and upper limits the minimum and maximum value in a data set must be determined.
Spreadsheets include functions to calculate the minimum value MIN and maximum value MAX in a data set.
=MIN(data)
=MAX(data)
In OpenOffice the MIN and MAX function can take a list of semi-colon separated numbers or a range of cells in a
spreadsheet. In statistics a range of cells is the most common input for these functions. When a range of cells is the
usual input, this text uses the word "data" to refer to the fact that the range of cells is usually your data! Ranges of cells
use two cell addresses separated by a full colon. An example is shown below where the data is arranged in a vertical
column from A2 to A42.
=MIN(A2:A42)
How to make a binned histogram at the ratio level
1. Find the minimum value of the data set using the MIN function in Excel
2. Find the maximum value of the data set using the MAX function in Excel
3. Calculate
the
range
by
subtracting
the
MIN
from
the
MAX:
range = maximum value - minimum value
4. Decide on the number of bins you are going to use (also called intervals or classes)
5. Divide the range by the number of bins to calculate the bin width (or interval width or class width)
6. Calculate the bin upper limit (also called class upper limit or interval upper limit)
7. Put the bin upper limits into a column of cells in Excel
8. Manually tally the data into the frequency column to determine the frequencies for each bin. The bin upper
limit is included in each tally.
9. Create a column chart
Bin Upper Limits (BUL) Frequency
=min + bin width
+ bin width
+ bin width
+ bin width
+ bin width = max
For the female height data:
58, 58, 59.5, 59.5, 60, 60, 60, 60, 60, 61, 61, 61.2, 61.5, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 63, 63, 63, 63.5,
64, 64, 64, 64, 65, 65, 66, 66
Five
bins
Min
Max
Range
=
Width = 8/5 = 1.6
would
produce
=
=
66
the
-
Calculation
58 + 1.6
59.6 + 1.6
61.2 + 1.6
62.8 + 1.6
64.4 + 1.6
following
58
=
results:
58
66
8
Height (BUL) Frequency
59.6
4
61.2
8
62.8
13
64.4
8
66
4
Sum:
37
Note that 61.2 is INCLUDED in the bin that ends at 61.2. The bin includes values at the bin upper limit. In other words,
a bin includes all values up to and including the bin upper limit. After making the column chart, double click on the
columns to open the data series dialog box. Find the Options tab and set the spacing (or gap width) to zero.
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Note that the spacing or gap width on the columns has been set to zero.
Relative Frequency
Relative frequency is one way to determine a probability.
Divide each frequency by the sum (the sample size) to get the relative frequency
Height BUL Frequency Relative Frequency f/n or P(x)
59.6
4
0.11
61.2
8
0.22
62.8
13
0.35
64.4
8
0.22
66
4
0.11
Sum:
37
1.00
The relative frequency always adds to one (rounding causes the above to add to 1.01, if all the decimal places were used
the relative frequencies would add to one.
The area under the relative frequency columns is equal to one.
Another example using integers:
0, 1, 2, 2, 3, 3, 3, 4, 4, 4, 4.5, 5, 5, 5, 6, 6, 7, 8, 9, 10
Five bins
min = 0
max = 10
range = 10
width = 10/5 = 2
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Bin Num
1
2
3
4
5>
Calculation
min + width
+ width
+ width
+ width
+ width
Bins Frequency Relative Frequency f/n or P(x)
2
4
0.20
4
6
0.30
6
6
0.30
8
2
0.10
10
2
0.10
Sum: 20
1.00
The above method produces equal width bins and to conforms the inclusion of the class upper limit by spreadsheet
packages.
Frequency function
For more advanced spreadsheet users, frequency data can be obtained using the frequency function FREQUENCY. This
function is also very useful when working with large data sets. In OpenOffice.org Calc the frequency function is:
=FREQUENCY(DATA;BINS)
In Microsoft Excel the function uses a comma instead of a semi-colon:
=FREQUENCY(DATA,BINS)
DATA refers to the range of cells containing the data, BINS refers to the range of cells containing the bin upper limits.
The data set seen below are the height measurements for 49 female students in statistics courses during two consecutive
terms.
The frequency function built into spreadsheets works very differently from all other functions. The frequency function
called an "array" function because the function places values into an array of cells. For the function to do this, you must
first select the cells into which the function will place the frequency values.
With the cells still highlighted, start typing the frequency function.
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After typing the opening parenthesis, drag and select the data to be binned. If the data is more than can be selected by
dragging, type the data range in by hand.
For OpenOffice.org Calc, type a semi-colon. For Microsoft Office Excel, type a comma.
Drag and select the bin upper limits.
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Type the closing parenthesis.
Then press and hold down BOTH the CONTROL (Ctrl) key and the SHIFT key. With both the control and shift keys
held down, press the Enter (or Return) key.
As noted above, the frequencies should add to the sample size. When working with spreadsheets, internal rounding
errors can cause the maximum value in a data set to not get included in the final bin. In the last bin, use the value
obtained by the MAX function and not the previous bin + a width formula to generate that bin upper limit.
2.3 Shapes of Distributions
The shapes of distributions have names by which they are known.
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One of the aspects of a sample that is often similar to the population is the shape of the distribution. If a good random
sample of sufficient size has a symmetric distribution, then the population is likely to have a symmetric distribution.
The process of projecting results from a sample to a population is called generalizing. Thus we can say that the shape
of a sample distribution generalizes to a population.
Endnote: Creating histograms with spreadsheets
Making histograms with OpenOffice.org Calc
Select both the column with the bins and the column with the frequencies.
- 20 -
Click on the chart wizard button and then drag the mouse to place and size the histogram.
At the first dialog box be sure to click on the "First column as label" check box as indicated by the arrow in the diagram
below.
For the next two screens simply click on "Next"
On the next screen fill in the appropriate titles. The legend can be "unchecked" as seen below.
- 21 -
When done, click on Create.
Double click any column to open up the data series dialog box.
- 22 -
Click on the options tab and set the spacing to zero.
Click on OK.
Making histograms with Microsoft Excel
Select ONLY the column with the column with the frequencies. Click on the chart wizard.
- 23 -
Click on next.
In step 2 of 4, click on the series tab
Click in the Category (X) axis labels text box
- 24 -
Select the bin upper limits by dragging with the mouse. Click on next when done.
Fill in the appropriate titles and then click on finish.
- 25 -
Double click any column to open up the Format Data series dialog box.
Click on the options tab and set the gap width to zero.
- 26 -
Click on OK.
- 27 -
Measures of Middle and Spread
3.1 Measures of central tendency: mode, median, mean, midrange
Mode
The mode is the value that occurs most frequently in the data. Spreadsheet programs such as Microsoft Excel or
OpenOffice.org Calc can determine the mode with the function MODE.
=MODE(data)
In the Fall of 2000 the statistics class gathered data on the number of siblings for each member of the class. One student
was an only child and had no siblings. One student had 13 brothers and sisters. The complete data set is as follows:
1,2,2,2,2,2,3,3,4,4,4,5,5,5,7,8,9,10,12,12,13
The mode is 2 because 2 occurs more often than any other value. Where there is a tie there is no mode.
For the ages of students in that class
18,19,19,20,20,21,21,21,21,22,22,22,22,23,23,24,24,25,25,26
...there is no mode: there is a tie between 21 and 22, hence there no single must frequent value. Spreadsheets will,
however, usually report a mode of 21 in this case. Spreadsheets often select the first mode in a multi-modal tie.
If all values appear only once, then there is no mode. Spreadsheets will display #N/A or #VALUE to indicate an error
has occurred - there is no mode. No mode is NOT the same as a mode of zero. A mode of zero means that zero is the
most frequent data value. Do not put the number 0 (zero) for "no mode." An example of a mode of zero might be the
number of children for students in statistics class.
Median
The median is the central (or middle) value in a data set. If a number sits at the middle, then it is the median. If the
middle is between two numbers, then the median is half way between the two middle numbers.
For the sibling data...
1,2,2,2,2,2,3,3,4,4,|4|,5,5,5,7,8,9,10,12,12,13
...the median is 4.
Note the data must be in order (sorted) before you can find the median. For the data 2, 4, 6, 8 the median is 5: (4+6)/2.
The median function in spreadsheets is MEDIAN.
=MEDIAN(data)
Mean (average)
The mean, also called the arithmetic mean and also called the average, is calculated mathematically by adding the
values and then dividing by the number of values (the sample size n).
If the mean is the mean of a population, then it is called the population mean μ. The letter μ is a Greek lower case "m"
and is pronounced "mu."
If the mean is the mean of a sample, then it is the sample mean x. The symbol x is pronounced "x bar."
The sum of the data ∑ x can be determined using the function =SUM(data). The sample size n can be determined using
=COUNT(data). Thus =SUM(data)/COUNT(data) will calculate the mean. There is also a single function that calculates
the mean. The function that directly calculates the mean is AVERAGE
=AVERAGE(data)
- 28 -
Resistant measures: One that is not influenced by extremely high or extremely low data values. The median tends to
be more resistant than mean.
Population mean and sample mean
If the mean is measured using the whole population then this would be the population mean. If the mean was calculated
from a sample then the mean is the sample mean. Mathematically there is no difference in the way the population and
sample mean are calculated.
Midrange
The midrange is the midway point between the minimum and the maximum in a set of data. The midrange is calculated
from:
midrange = (maximum + minimum)/2
In a spreadsheet use the following formula:
=(MAX(data)+MIN(data))/2
3.2 Differences in the Distribution of Data
Range
Consider the following data:
Data
mode
median
mean μ
min max
range
midrange
Data set 1 5, 5, 5, 5
5
5
5
5
5
0
0
Data set 2 2, 4, 6, 8
none
5
5
2
8
6
3
Data set 3 2, 2, 8, 8
none
5
5
2
8
6
3
Neither the mode, median, nor the mean reveal clearly the differences in the distribution of the data above. The mean
and the median are the same for each data set. The mode is the same as the mean and the median for the first data set
and is unavailable for the last data set (spreadsheets will report a mode of 2 for the last data set). A single number that
would characterize how much the data is spread out would be useful.
The range is one way to capture the spread of the data. The range is calculated by subtracting the smallest value from
the largest value. In a spreadsheet:
=MAX(data) − MIN(data)
The range still does not characterize the difference between set 2 and 3: the last set has more data further away from the
center of the data distribution. The range misses this difference.
To capture the spread of the data we use a measure related to the average distance of the data from the mean. We call
this the standard deviation. If we have a population, we report this average distance as the population standard
deviation. If we have a sample, then our average distance value may underestimate the actual population standard
deviation. As a result the formula for sample standard deviation adjusts the result mathematically to be slightly larger.
For our purposes these numbers are calculated using spreadsheet functions.
Standard deviation
One way to distinguish the difference in the distribution of the numbers in data set 2 and data set 3 above is to use the
standard deviation.
Data
Data set 1
5, 5, 5, 5
Data set 2
2, 4, 6, 8
Data set 3
2, 2, 8, 8
The function that calculates the sample standard deviation is:
mean μ
5
5
5
stdev
0.00
2.58
3.46
=STDEV(data)
In
this
text
the
symbol
for
the
sample
standard
In this text the symbol for the population standard deviation is usually σ.
deviation
is
usually
sx.
The symbol sx usually refers the standard deviation of single variable x data. If there is y data, the standard deviation of
the y data is sy. Other symbols that are used for standard deviation include s and σx. Some calculators use the unusual
and confusing notations σxn−1 and σxn for sample and population standard deviations.
- 29 -
In this class we always use the sample standard deviation in our calculations. The sample standard deviation is
calculated in a way such that the sample standard deviation is slightly larger than the result of the formula for the
population standard deviation. This adjustment is needed because a population tends to have a slightly larger spread
than a sample. There is a greater probability of outliers in the population data.
Coefficient of variation CV
The Coefficient of Variation is calculated by dividing the standard deviation (usually the sample standard deviation) by
the mean.
=STDEV(data)/AVERAGE(data)
Note that the CV can be expressed as a percentage: Group 2 has a CV of 52% while group 3 has a CV of 69%. A
deviation of 3.46 is large for a mean of 5 (3.46/5 = 69%) but would be small if the mean were 50 (3.46/50 = 7%). So the
CV can tell us how important the standard deviation is relative to the mean.
Rules of thumb regarding spread
As an approximation, the standard deviation for data that has a symmetrical, heap-like distribution is roughly onequarter of the range. If given only minimum and maximum values for data, this rule of thumb can be used to estimate
the standard deviation.
At least 75% of the data will be within two standard deviations of the mean, regardless of the shape of the distribution
of
the
data.
At least 89% of the data will be within three standard deviations of the mean, regardless of the shape of the distribution
of
the
data.
If the shape of the distribution of the data is a symmetrical heap, then as much as 95% of the data will be within two
standard deviation of the mean.
Data beyond two standard deviations away from the mean is considered "unusual" data.
Basic statistics and their interaction with the levels of measurement
Levels of measurement and appropriate measures
Level of measurement Appropriate measure of middle Appropriate measure of spread
nominal
mode
none or number of categories
ordinal
median
range
interval
median or mean
range or standard deviation
ratio
mean
standard deviation
At the interval level of measurement either the median or mean may be more appropriate depending on the specific
system being studied. If the median is more appropriate, then the range should be quoted as a measure of the spread of
the data. If the mean is more appropriate, then the standard deviation should be used as a measure of the spread of the
data.
Another way to understand the levels at which a particular type of measurement can be made is shown in the following
table.
Levels at which a particular statistic or parameter has meaning
Level of measurement
Nominal Ordinal Interval
Ratio
sample size
mode
minimum
maximum
Statistic/
Parameter
range
median
mean
standard deviation
coefficient of variation
For example, a mode, median, and mean can be calculated for ratio level measures. Of those, the mean is usually
considered the best measure of the middle for a random sample of ratio level data.
- 30 -
3.3 Variables
Discrete Variables
When there are a countable number of values that result from observations, we say the variable producing the results is
discrete. The nominal and ordinal levels of measurement almost always measure a discrete variable.
The following examples are typical values for discrete variables:
•
•
•
true or false (2 values)
yes or no (2 values)
strongly agree | agree | neutral | disagree | strongly disagree (5 values)
The last example above is a typical result of a type of survey called a Likert survey developed by Renis Likert in 1932.
When reporting the "middle value" for a discrete distribution at the ordinal level it is usually more appropriate to report
the median. For further reading on the matter of using mean values with discrete distributions refer to the pages by
Nora Mogey and by the Canadian Psychiatric Association.
Note that if the variable measures only the nominal level of measurement, then only the mode is likely to have any
statistical "meaning", the nominal level of measurement has no "middle" per se.
There may be rare instances in which looking at the mean value and standard deviation is useful for looking at
comparative performance, but it is not a recommended practice to use the mean and standard deviation on a discrete
distribution. The Canadian Psychiatric Association discusses when one may be able to "break" the rules and calculate a
mean on a discrete distribution. Even then, bear in mind that ratios between means have no "meaning!"
For example, questionnaire's often generate discrete results:
Never
0
About once About once A few times Every
a month
a week
a week
day
1
2
3
4
How often do you drink caffeinated drinks such as coffee,
tea, or cola?
How often do you chew tobacco without betelnut?
How often do you chew betelnut without tobacco?
How often do you chew betelnut with tobacco?
How often do you drink sakau en Pohnpei?
How often do you drink beer?
How often do you drink wine?
How often do you drink hard liquor
(whisky, rum, vodka, tequila, etc.)?
How often do you smoke cigarettes?
How often do you smoke marijuana?
How often do you use controlled substances other than
marijuana? (methamphetamines, cocaine, crack, ice, shabu,
etc.)?
The results of such a questionnaire are numeric values from 0 to 4. For an example of a real student alcohol
questionnaire, see: http://www.indiana.edu/~engs/saq.html
Continuous Variables
When there is a infinite (or uncountable) number of values that may result from observations, we say that the variable is
continuous. Physical measurements such as height, weight, speed, and mass, are considered continuous measurements.
Bear in mind that our measurement device might be accurate to only a certain number of decimal places. The variable is
continuous because better measuring devices should produce more accurate results.
The following examples are continuous variables:
•
•
•
•
•
•
•
distance
time
mass
length
height
depth
weight
- 31 -
•
•
speed
body fat
When reporting the "middle value" for a continuous distribution it is appropriate to report the mean and standard
deviation. The mean and standard deviation only have "meaning" for the ratio level of measurement.
Interactions between levels of measure, variable type, and measures of middle and
spread
Level of
measurement
nominal
ordinal
ratio
Levels of measurement and appropriate measures
Typical variable
Appropriate measure of
Appropriate measure of middle
type
variation
discrete
mode
none
discrete
median (can also report mode)
range
mean (can also report median and
continuous
sample standard deviation
mode)
3.4 A Measure of Relative Standing
Z-scores are a useful way to combine scores from data that has different means and standard deviations. Z-scores are an
application of the above measures of center and spread.
Remember that the mean is the result of adding all of the values in the data set and then dividing by the number of
values in the data set. The word mean and average are used interchangeably in statistics.
Recall also that the standard deviation can be thought of as a mathematical calculation of the average distance of the
data from the mean of the data. Note that although I use the words average and mean, the sentence could also be written
"the mean distance of the data from the mean of the data."
Z-Scores
Z-scores simply indicate how many standard deviations away from the mean is a particular score. This is termed
"relative standing" as it is a measure of where in the data the score is relative to the mean and "standardized" by the
standard deviation. The formula for z is:
Note the parentheses!
Data that is two standard deviations below the mean will have a z-score of −2, data that is two standard deviations
above the mean will have a z-score of +2. Data beyond two standard deviations away from the mean will have z-scores
below −2 or above 2.
Why z-scores?
Suppose a test has a mean score of 10 and a standard deviation of 2 with a total possible of 20. Suppose a second test
has the same mean of 10 and total possible of 20 but a standard deviation of 8.
On the first test a score of 18 would be rare, an unusual score. On the first test 89% of the students would have scored
between 6 and 16 (three standard deviations below the mean and three standard deviations above the mean.
- 32 -
On the second test a score of 18 would only be one standard deviation above the mean. This would not be unusual, the
second test had more spread.
Adding two scores of 18 and saying the student had a score of 36 out of 40 devalues what is a phenomenal performance
on the first test.
Converting to z-scores, the relative strength of the performance on test one is valued more strongly. The z-score on test
one would be (18-10)/2 = 4, while on test two the z-score would be (18-10)/8 = 1. The unusually outstanding
performance on test one is now reflected in the sum of the z-scores where the first test contributes a sum of 4 and the
second test contributes a sum of 1.
When values are converted to z-scores, the mean of the z-scores is zero. A student who scored a 10 on either of the tests
above would have a z-score of 0. In the world of z-scores, a zero is average!
Z-scores also adjust for different means due to differing total possible points on different tests.
Consider again the first test that had a mean score of 10 and a standard deviation of 2 with a total possible of 20. Now
consider a third test with a mean of 100 and standard deviation of 40 with a total possible of 200. On this third test a
score of 140 would be high, but not unusually high.
Adding the scores and saying the student had a score of 158 out of 220 again devalues what is a phenomenal
performance on test one. The score on test one is dwarfed by the total possible on test three. Put another way, the 18
points of test one are contributing only 11% of the 158 score. The other 89% is the test three score. We are giving an
eight-fold greater weight to test three. The z-scores of 4 and 1 would add to five. This gives equal weight to each test
and the resulting sum of the z-scores reflects the strong performance on test one with an equal weight to the ordinary
performance on test three.
Z-scores only provide the relative standing. If a test is given again and all students who take the test do better the second
time, then the mean rises and like a tide "lifts all the boats equally." Thus an individual student might do better, but
because the mean rose, their z-score could remain the same. This is also the downside to using z-scores to compare
performances between tests - changes in "sea level" are obscured. One would have to know the mean and standard
deviation and whether they changed to properly interpret a z-score.
- 33 -
Paired Data and Scatter Diagrams
4.1 Best Fit Lines: Linear Regressions
A runner runs from the College of Micronesia-FSM National campus to PICS via the powerplant/Nahnpohnmal back
road. The runner tracks his time and distance.
Location
Time x (minutes) Distance y (km)
College
0
0
Dolon Pass
20
3.3
Turn-off for Nahnpohnmal 25
4.5
Bottom of the beast
33
5.7
Top of the beast
34.5
5.9
Track West
55
9.7
PICS
56
10.1
Is there a relationship between the time and the distance? If there is a relationship, then data will fall in a patterned
fashion on an xy graph. If there is no relationship, then there will be no shape to the pattern of the data on a graph.
If the relationship is linear, then the data will fall roughly along a line. Plotting the above data yields the following
graph:
The data falls roughly along a line, the relationship appears to linear. If we can find the equation of a line through the
data, then we can use the equation to predict how long it will take the runner to cover distances not included in the table
above, such as five kilometers. In the next image a best fit line has been added to the graph.
The best fit line is also called the least squares line because the mathematical process for determining the line
minimizes the square of the vertical displacement of the data points from the line. The process of determining the best
fit line is also known and performing a linear regression. Sometimes the line is referred to as a linear regression.
The graph of time versus distance for a runner is a line because a runner runs at the same pace kilometer after kilometer.
- 34 -
4.2 Slope and Intercept
Slope
A spreadsheet is used to find the slope and the y-intercept of the best fit line through the data.
To get the slope m with OpenOffice Calc use the function:
=SLOPE(y-values;x-values)
Note that the y-values are entered first, the x-values are entered second. This is the reverse of traditional algebraic order
where coordinate pairs are listed in the order (x, y). The x and y-values are usually arranged in columns. The column
containing the x data is usually to the left of the column containing the y-values. An example where the data is in the
first two columns from row two to forty-two can be seen below.
=SLOPE(B2:B42;A2:A42)
Intercept
The intercept is usually the starting value for a function. Often this is the y data value at time zero, or distance zero.
To get the intercept:
=INTERCEPT(y-values;x-values)
Note that intercept also reverses the order of the x and y values!
For the runner data above the equation is:
distance = slope * time + y-intercept
distance = 0.18 * time + − 0.13
y = 0.18 * x
+ − 0.13
or
y = 0.18x − 0.13
where x is the time and y is the distance
In algebra the equation of a line is written as y = m*x + b where m is the slope and b is the intercept. In statistics the
equation of a line is written as y = a + b*x where a is the intercept (the starting value) and b is the slope. The two fields
have their own traditions, and the letters used for slope and intercept are a tradition that differs between the field of
mathematics and the field of statistics.
Using the y = mx + b equation we make predictions about how far the runner will travel given a time, or how long the
runner will runner given a distance. For example, according the equation above, a 45 minute run will result in the runner
covering 0.18*45 - 0.13 = 7.97 kilometers. Using the inverse of the equation we can predict that the runner can run a
five kilometer distance in 28.5 minutes (28 minutes and 30 seconds).
Given any time, we can calculate the distance. Given any distance, can solve for the time.
Microsoft Excel can be used to create a chart that displays the linear regression trendline. The formulas for Excel are
almost the same as for OpenOffice Calc with one difference: Excel uses a comma instead of a semi-colon.
=SLOPE(y-values,x-values)
=INTERCEPT(y-values,x-values)
4.25 Creating an xy scattergraph using OpenOffice.org Calc 2.4,
3.0
The data used in the following examples is contained in the following table.
Evening joggle location Time x (min) Distance y (m)
Dolihner
0.0
0
Pohnpei campus
9.0
1250
Mesenieng outbound
16.7
2600
Mesenieng inbound
26.6
4200
Pwunso botanic
35.7
5300
Dolihner
41.9
6190
Making an xy scattergraph using OpenOffice.org 2.4. The first portion of these directions apply to both OpenOffice.org
Calc 2.3, 2.4, and 3.0.
- 35 -
First select the data to be graphed.
Then click on the chart wizard button
in the toolbar to start the chart wizard.
Choose an XY (Scatter) graph in the first dialog box.
For statistics class, click through the next two dialog boxes.
- 36 -
In the fourth and final dialog box you can set up the x and y axis labels as well as a chart title. Then click on Finish.
Before clicking anywhere else, choose Insert: Statistics from the menu.
Click on the linear regression option in the Regression curves boxes on the right side of the dialog box.
- 37 -
Click on OK to close the statistics dialog box.
This next section applies to OpenOffice.org Calc version 2.4
Click on the finished chart, and roll the tip of the mouse cursor arrow onto the linear regression line. This will take
some practice and precision in pointing. You are on the linear regression line when you see the Regression Curve tool
tip.
Right click when you see the Regression Curve tool tip to open the context menu.
Choose Object Properties to open the regression curve dialog box. Note that one can directly insert the regression curve
from the right-click context menu, but the correlation coefficient R2 can only be accessed from the dialog box in 2.4.
- 38 -
Choose the Equation tab and click in the check boxes for show equation and show correlation coefficient (R2).
Click OK in the dialog box and the equation along with R2 is displayed on the graph. R2 is displayed rounded to two
decimal places.
- 39 -
OpenOffice.org 3.0
OpenOffice.org 3.0 integrates the linear regression options into a single Trend Lines item on the Insert menu. After
clicking on Finish in the chart wizard dialog series, choose Trend Lines from the Insert menu.
The Trend Lines dialog box permits selection of linear and non-linear regression lines. For a straight line regression,
choose the linear regression type. Click on the check boxes for the function and the R 2. You may have to click twice to
obtain the check mark.
Click on OK to close the dialog box.
All versions of OpenOffice.org Calc can calculate the slope and intercept using spreadsheet functions.
Specifics for creating XY scatterplots in Microsoft Excel and OpenOffice Calc 2.0 can be seen at:
http://www.comfsm.fm/~dleeling/statistics/xyscattergraph.html
4.3 Relationships between variables
After plotting the x and y data, the xy scattergraph helps determine the nature of the relationship between the x values
and the y values. If the points lie along a straight line, then the relationship is linear. If the points form a smooth curve,
then the relationship is non-linear (not a line). If the points form no pattern then the relationship is random.
- 40 -
Relationships between two sets of data can be positive: the larger x gets, the larger y gets.
Relationships between two sets of data can be negative: the larger x gets, the smaller y gets.
Relationships
between
two
sets
of
data
can
be
non-linear
Relationships between two sets of data can be random: no relationship exists!
For the runner data above, the relationship is a positive relationship. The points line along a line, therefore the
relationship is linear.
An example of a negative relationship would be the number of beers consumed by a student and a measure of the
physical coordination. The more beers consumed the less their coordination!
4.4 Correlation
For a linear relationship, the closer to a straight line the points fall, the stronger the relationship. The measurement that
describes how closely to a line are the points is called the correlation.
The following example explores the correlation between the distance of a business from a city center versus the amount
of product sold per person. In this case the business are places that serve pounded Piper methysticum plant roots, known
elsewhere as kava but known locally as sakau. This business is unique in that customers self-limit their purchases,
buying only as many cups of sakau as necessary to get the warm, sleepy, feeling that the drink induces. The businesses
are locally referred to as sakau markets. The local theory is that the further one travels from the main town (and thus
deeper into the countryside of Pohnpei) the stronger the sakau that is served. If this is the case, then the mean number of
cups should fall with distance from the main town on the island.
The following table uses actual data collected from these businesses, the names of the businesses have been changed.
Sakau Market distance/km (x) mean cups per person (y)
Upon the river 3.0
5.18
Try me first
13.5
3.93
At the bend
14.0
3.19
Falling down 15.5
2.62
The first question a statistician would ask is whether there is a relationship between the distance and mean cup data.
Determining whether there is a relationship is best seen in an xy scattergraph of the data.
If we plot the points on an xy graph using a spreadsheet, the y-values can be seen to fall with increasing x-value. The
- 41 -
data points, while not all exactly on one line, are not far away from the best fit line. The best fit line indicates a negative
relationship. The larger the distance, the smaller the mean number of cups consumed.
We use a number called the Pearson product-moment correlation coefficient r to tell us how well the data fits to a
straight line. The full name is long, in statistics this number is called simply r. R can be calculated using a spreadsheet
function.
The OpenOffice Calc and Microsoft Excel function for calculating r are similar. The OpenOffice Calc function is:
=CORREL(y-values;x-values)
For Microsoft Excel the function for correlation is the same, only the semi-colon changes to a comma:
=CORREL(y-values,x-values)
The Pearson product-moment correlation coefficient r (or just correlation r) values that result from the formula are
always between -1 and 1. One is perfect positive linear correlation. Negative one is perfect negative linear correlation. If
the correlation is zero or close to zero: no linear relationship between the variables.
A guideline to r values:
- 42 -
Note that perfect has to be perfect: 0.99999 is very close, but not perfect. In real world systems perfect correlation,
positive or negative, is rarely or never seen. A correlation of 0.0000 is also rare. Systems that are purely random are also
rarely seen in the real world.
Spreadsheets usually round to two decimals when displaying decimal numbers. A correlation r of 0.999 is displayed as
"1" by spreadsheets. Use the the Format menu to select the cells item. In the cells dialog box, click on the numbers tab
to increase the number of decimal places. When the correlation is not perfect, adjust the decimal display and write out
all the decimals.
The correlation r of − 0.93 is a strong negative correlation. The relationship is strong and the relationship is negative.
The equation of the best fit line, y = −0.18x + 5.8 where y is the mean number of cups and x is the distance from the
main town. The equations that generated the slope, y-intercept, and correlation can be seen in the earlier image.
The strong relationship means that the equation can be used to predict mean cup values, at least for distances between
3.0 and 15.5 kilometers from town.
A second example explores the correlation between female students pounds of fat in a statistics course. The table
provides the x and y data.
The table below continues beyond the boundaries of the page.
Age/years (x)
18
18
19
19
19
19
19
19
19
19
19
19
19
19
19
19
20
20
20
20
20
Fat/pounds (y) 29.5 51.6 14.9 25.7 26.4 29.0 29.8 29.9 30.8 30.9 33.5 35.1 36.2 36.8 37.0 69.0 20.6 27.7 28.2 33.2 33.4
Age/years (x)
20
20
20
20
20
20
20
20
20
20
21
21
21
21
22
22
22
22
23
23
Fat/pounds (y) 36.5 39.3 39.4 40.2 48.9 50.4 56.7 57.8 59.7 107.8 22.7 34.2 65.0 76.9 28.1 34.8 37.2 38.3 28.0 46.8
The first question a statistician would ask is whether there is a relationship seen in the xy scattergraph between the age
of a female student at COMFSM and the pounds of fat? Can we use our data to predict a pounds of body fat based on
age alone?
If we plot the points on an xy graph using a spreadsheet, the data does not appear to be strongly linear. The data appears
to be scattered randomly about the graph. Although a spreadsheet is able to give us a best fit line (a linear regression or
least squares line), we will later have to consider whether the relationship is strong enough to make the equation useful.
- 43 -
In the example above the correlation r is 0.09. Zero would be random correlation. This value is so close to zero that the
correlation is effectively random. The relationship is random. There is no relationship. The linear equation y = 1.21x +
15.68, where y is the pounds of fat and x is the age, cannot be used to predict the pounds of fat given the age.
Limitations of linear regressions
We cannot usually predict values that are the minimum or maximum x values and make meaningful predictions. In the
example of the runner, we could calculate how far the runner would run in 72 hours (three days and three nights) but it
is unlikely the runner could run continuously for that length of time. For some systems values can be predicted below
the minimum x or above the maximum x value. When we do this it is called extrapolation. Very few systems can be
extrapolated, but some systems remain linear for values near to the provided x values.
Coefficient of Determination r2
The coefficient of determination, r2, is a measure of how much of the variation in the independent x variable explains
the variation in the dependent y variable. This does NOT imply causation. In spreadsheets the ^ symbol (shift-6) is
exponentiation. In OpenOffice Calc we can square the correlation with the following formula:
=(CORREL(y-values;x-values))^2
The result, which is between 0 and 1 inclusive, is often expressed as a percentage.
Imagine a Yamaha outboard motor fishing boat sitting out beyond the reef in an open ocean swell. The swell moves the
boat gently up and down. Now suppose there is a small boy moving around in the boat. The boat is rocked and swayed
by the boy. The total motion of the boat is in part due to the swell and in part due to the boy. Maybe the swell accounts
for 70% of the boat's motion while the boy accounts for 30% of the motion. A model of the boat's motion that took into
account only the motion of the ocean would generate a coefficient of determination of about 70%.
Causality
Finding that a correlation exists does not mean that the x-values cause the y-values. A line does not imply causation:
Your age does not cause your pounds of body fat, nor does time cause distance for the runner.
Studies in the mid 1800s of Micronesia would have shown of increase each year in church attendance and sexually
transmitted diseases (STDs). That does NOT mean churches cause STDs! What the data is revealing is a common
variable underlying our data: foreigners brought both STDs and churches. Any correlation is simply the result of the
common impact of the increasing influence of foreigners.
- 44 -
Calculator usage notes
Some calculators will generate a best fit line. Be careful. In algebra straight lines had the form y = mx + b where m was
the slope and b was the y-intercept. In statistics lines are described using the equation y = a + bx. Thus b is the slope!
And a is the y-intercept! You would not need to know this but your calculator will likely use b for the slope and a for
the y-intercept. The exception is some TI calculators that use SLP and INT for slope and intercept respectively.
Physical science note
Note only for those in physical science courses. In some physical systems the data point (0,0) is the most accurately
known measurement in a system. In this situation the physicist may choose to force the linear regression through the
origin at (0,0). This forces the line to have an intercept of zero. There is another function in spreadsheets which can
force the intercept to be zero, the LINear ESTimator function. The following functions use time versus distance,
common x and y values in physical science.
=LINEST(distance (y) values;time (x) values;0)
The formulas for Microsoft Excel are almost the same as for OpenOffice Calc with one difference: Excel uses a comma
instead of a semi-colon.
=LINEST(distance (y) values,time (x) values,0)
Note that the same as the slope and intercept functions, the y-values are entered first, the x-values are entered second.
- 45 -
Introduction to Probability
5.1 Ways to determine a probability
A probability is the likelihood of an event or outcome. Probabilities are specified mathematically by a number between
0 and 1 including 0 or 1.
•
•
•
•
0 is no likelihood an event will occur.
1 is absolute certainty an event will occur.
0.5 is an equal likelihood of occurrence or non-occurrence.
Any value between 0 and 1 can occur.
We use the notation P(eventLabel) = probability to report a probability.
There are three ways to assign probabilities.
1. Intuition or subjective estimate
2. Equally likely outcomes
3. Relative Frequencies
Intuition
Intuition/subjective measure. An educated best guess. Using available information to make a best estimate of a
probability. Could be anything from a wild guess to an educated and informed estimate by experts in the field.
Equally Likely Events or Outcomes
Equally Likely Events: Probabilities from mathematical formulas
In the following the word "event" and the word "outcome" are taken to have the same meaning.
Probabilities versus Statistics
The study of problems with equally likely outcomes is termed the study of probabilities. This is the realm of the
mathematics of probability. Using the mathematics of probability, the outcomes can be determined ahead of time.
Mathematical formulas determine the probability of a particular outomce. All measures are population parameters. The
mathematics of probability determines the probabilities for coin tosses, dice, cards, lotteries, bingo, and other games of
chance.
This course focuses not on probability but rather on statistics. In statistics, measurement are made on a sample taken
from the population and used to estimate the population's parameters. All possible outcomes are not usually known. is
usually not known and might not be knowable. Relative frequencies will be used to estimate population parameters.
Calculating Probabilities
Where each and every event is equally likely, the probability of an event occurring can be determined from
probability = ways to get the desired event/total possible events
or
probability = ways to get the particular outcome/total possible outcomes
Dice and Coins
Binary probabilities: yes or no, up or down, heads or tails
A penny
P(head on a penny) = one way to get a head/two sides = 1/2 = 0.5 or 50%
That probability, 0.5, is the probability of getting a heads or tails prior to the toss. Once the toss is done, the coin is
either a head or a tail, 1 or 0, all or nothing. There is no 0.5 probability anymore.
Over any ten tosses there is no guarantee of five heads and five tails: probability does not work like that. Over any small
sample the ratios of expected outcomes can differ from the mathematically calculated ratios.
- 46 -
Over thousands of tosses, however, the ratio of outcomes such as the number of heads to the number of tails, will
approach the mathematically predicted amount. We refer to this as the law of large numbers.
In effect, a few tosses is a sample from a population that consists, theoretically, of an infinite number of tosses. Thus we
can speak about a population mean μ for an infinite number of tosses. That population mean μ is the mathematically
predicted probability.
Population mean μ = (number of ways to get a desired outcome)/(total possible outcomes)
Dice: Six-sided
A six-sided die. Six sides. Each side equally likely to appear. Six total possible outcomes. Only one way to roll a one:
the side with a single pip must face up. 1 way to get a one/6 possible outcomes = 0.1667 or 17%
P(1) = 0.17
Dice: Four, eight, twelve, and twenty sided
The formula remains the same: the number of possible ways to get a particular roll divided by the number of possible
outcomes (that is, the number of sides!).
Think about this: what would a three sided die look like? How about a two-sided die? What about a one sided die? What
shape would that be? Is there such a thing?
Two dice
Ways to get a five on two dice: 1 + 4 = 5, 2 + 3 = 5, 3 + 2 = 5, 4 + 1 = 5 (each die is unique). Four ways to get/36 total
possibilities = 4/36 = 0.11 or 11%
Homework:
1. What is the probability of rolling a three on...
1. A four sided die?
2. A six sided die?
3. An eight sided die?
4. A twelve sided die?
5. A twenty sided die labeled 0-9 twice.
2. What is the probability of throwing two pennies and having both come up heads?
5.2 Sample space
The sample space set of all possible outcomes in an experiment or system.
Bear in mind that the following is an oversimplification of the complex biogenetics of achromatopsia for the sake of a
statistics example. Achromatopsia is controlled by a pair of genes, one from the mother and one from the father. A child
is born an achromat when the child inherits a recessive gene from both the mother and father.
A is the dominant gene
a is the recessive gene
A person with the combination AA is "double dominant" and has "normal" vision.
A person with the combination Aa is termed a carrier and has "normal" vision.
A person with the combination aa has achromatopsia.
Suppose two carriers, Aa, marry and have children. The sample space for this situation is as follows:
mother
\ A a
father A AA Aa
a Aa aa
The above diagram of all four possible outcomes represents the sample space for this exercise. Note that for each and
every child there is only one possible outcome. The outcomes are said to be mutually exclusive and independent. Each
outcome is as likely as any other individual outcome. All possible outcomes can be calculated. the sample space is
completely known. Therefore the above involves probability and not statistics.
The probability of these two parents bearing a child with achromatopsia is:
P(achromat) = one way for the child to inherit aa/four possible combinations = 1/4 = 0.25 or 25%
- 47 -
This does NOT mean one in every four children will necessarily be an achromat. Suppose they have eight children.
While it could turn out that exactly two children (25%) would have achromatopsia, other likely results are a single child
with achromatopsia or three children with achromatopsia. Less likely, but possible, would be results of no achromat
children or four achromat children. If we decide to work from actual results and build a frequency table, then we would
be dealing with statistics.
The probability of bearing a carrier is:
P(carrier) = two ways for the child to inherit Aa/four possible combinations = 2/4 = 0.50
Note that while each outcome is equally likely,there are TWO ways to get a carrier, which results in a 50% probability
of a child being a carrier.
At your desk: mate an achromat aa father and carrier mother Aa.
1. What is the probability a child will be born an achromat? P(achromat) = ________
2. What is the probability a child will be born with "normal" vision? P("normal") = ______
Homework: Mate a AA father and an achromat aa mother.
1. What is the probability a child will be born an achromat? P(achromat) = ________
2. What is the probability a child will be born with "normal" vision? P("normal") = ______
See: http://www.achromat.org/ for more information on achromatopsia.
Genetically linked schizophrenia is another genetic example:
Mol Psychiatry. 2003 Jul;8(7):695-705, 643. Genome-wide scan in a large complex pedigree with
predominantly male schizophrenics from the island of Kosrae: evidence for linkage to chromosome
2q. Wijsman EM, Rosenthal EA, Hall D, Blundell ML, Sobin C, Heath SC, Williams R, Brownstein MJ,
Gogos JA, Karayiorgou M. Division of Medical Genetics, Department of Medicine, University of
Washington, Seattle, WA, USA. It is widely accepted that founder populations hold promise for mapping
loci for complex traits. However, the outcome of these mapping efforts will most likely depend on the
individual demographic characteristics and historical circumstances surrounding the founding of a given
genetic isolate. The 'ideal' features of a founder population are currently unknown. The Micronesian
islandic population of Kosrae, one of the four islands comprising the Federated States of Micronesia
(FSM), was founded by a small number of settlers and went through a secondary genetic 'bottleneck' in
the mid-19th century. The potential for reduced etiological (genetic and environmental) heterogeneity, as
well as the opportunity to ascertain extended and statistically powerful pedigrees makes the Kosraen
population attractive for mapping schizophrenia susceptibility genes. Our exhaustive case ascertainment
from this islandic population identified 32 patients who met DSM-IV criteria for schizophrenia or
schizoaffective disorder. Three of these were siblings in one nuclear family, and 27 were from a single
large and complex schizophrenia kindred that includes a total of 251 individuals. One of the most startling
findings in our ascertained sample was the great difference in male and female disease rates. A genomewide scan provided initial suggestive evidence for linkage to markers on chromosomes 1, 2, 3, 7, 13, 15,
19, and X. Follow-up multipoint analyses gave additional support for a region on 2q37 that includes a
schizophrenia locus previously identified in another small genetic isolate, with a well-established recent
genealogical history and a small number of founders, located on the eastern border of Finland. In addition
to providing further support for a schizophrenia susceptibility locus at 2q37, our results highlight the
analytic challenges associated with extremely large and complex pedigrees, as well as the limitations
associated with genetic studies of complex traits in small islandic populations. PMID: 12874606
[PubMed - indexed for MEDLINE]
The above article is both fascinating and, at the same time, calls into question privacy issues. On the small island of
Kosrae "three siblings from one nuclear family" are identifiable people.
5.3 Relative Frequency
The third way to assign probabilities is from relative frequencies. Each relative frequency represents a probability of
that event occurring for that sample space. Body fat percentage data was gathered from 58 females here at the College
since summer 2001. The data had the following characteristics:
count 59
mean 28.7
sx 7.1
min 15.6
max 50.1
A five bin (class) frequency and relative frequency table has the following results:
BFI = Body Fat Index (percentage*100)
- 48 -
CLL = Class (bin) Lower Limit
CUL = Class (bin) Upper Limit (Excel uses)
Note that the bins are not equal width.
BFI fem CUL Frequency
x
f
Athletically fit*
20
3
Physically fit
24
15
Acceptable
31
24
Borderline obese (overfat) 39
12
Medically obese
51
5
Sample size n: 59
* body fat percentage category
Medical Category
Relative Frequency
f/n or P(x)
0.05
0.25
0.41
0.20
0.08
1.00
This means there is a...
•
•
•
•
•
0.05 (five percent) probability of a female student in the sample having a body fat percentage between 12 and
20 (athletically fit)
0.25 (25%) probability of a female student in the sample has body fat percentage between 20.1 (the Tanita unit
only measured to the nearest tenth) and 24 (physically fit)
0.41 (41%) probability of a female student in the sample has body fat percentage between 24.1 and 31
(acceptable but not fit level of fat)
0.20 (20%) probability of a female student in the sample has body fat percentage between 31.1 and 39 (on the
borderline between acceptable and obese)
0.08 (8%) probability of a female student in the sample has body fat percentage between 39.1 and 51
(medically obese)
The most probable result (most likely) is a body fat measurement between 24.1 and 31 with a 41% probability of a
student being in each of either of these intervals.
The same table, but for male students:
BFI male CUL
x
Athletically fit*
13
Physically fit
17
Acceptable
20
Borderline obese (overfat) 25
Medically obese
50
Sample size n:
Medical Category
Frequency
f
9
11
10
9
12
Relative Frequency
f/n or P(x)
0.18
0.22
0.20
0.18
0.24
51
1.00
The male students have a higher probability of being obese than the female students!
Kosraens abroad: Another example
What is the probability that a Kosraen lives outside of Kosrae? An informal survey done on the 25th of December 2007
produced the following data. The table also includes data gathered Christmas 2003.
Location
Ebeye
Guam
Honolulu
Kona
Maui
Pohnpei
Seattle
Texas
Virgina Beach
USA Other
Diaspora sums:
2003 Conservative
200
600
200
100
200
200
200
200
1700
Kosraen population estimates
2003 Possible 2007 Growth
30
300
300 50%
1000
1000 67%
200
800 300%
100
60
-40%
200
300 50%
200
600 200%
200
N/A 200
N/A 200
N/A 2400
3090 - 49 -
Kosrae
7663
7663
8183 Est. Total Pop.: 9363
10063
11273 Percentage abroad: 18.2%
23.8%
27% 48%
The relative frequency of 27% is a point estimate for the probability that a Kosraen lives outside of Kosrae.
Law of Large Numbers
For relative frequency probability calculations, as the sample size increases the probabilities get closer and closer to the
true population parameter (the actual probability for the population). Bigger samples are more accurate.
5.4 Or
Probabilities can add. The probability that a female student is either athletically fit, physically fit, acceptable, or
borderline can be calculated by adding the probabilities
P(females students are athletically fit OR physically fit OR acceptable OR borderline) = 0.05 + 0.25 + 0.41 + 0.20 =
0.91
Note that each student has one and only one body fat measurement, the outcomes are independent and mutually
exclusive. When the outcomes are independent the probabilities add when the word OR is used.
P(A or B) = P(A) + P(B)
And
For mutually exclusive and independent events, the probability that event A and event B will occur is calculated by
multiplying the individual probabilities. However, this has no clear meaning in the above context. A student cannot be
athletically fit and medically obese at the same time.
Complement of an Event (not compliment!)
The complement of an event is the probability that the event will not occur. Since all probabilities add to one, the
complement can be calculated from 1 - P(x). The complement is sometimes written P(NOT event). In the foregoing
example we calculated P(Not medically obese) = 0.91
Non-mutually exclusive outcomes/dependent outcomes
Consider the following table of unofficial results from the summer 2000 senatorial election in Kitti and Madolehnihmw.
Candidates from both Kitti and Madolehnihmw ran for office. One Kitti candidate was advised that he was spending too
much time in Madolehnihmw, that he would not draw a lot of votes from Madolehnihmw. To what extent, if any, is this
true? Can we determine the "loyalty" of the voters and make a determination as to whether campaigning outside one's
home municipality matters?
K
M
M
K
M
K
K
M
M
DEdwa BEtse BHelg OILawr DGNeth STSalv HSeme JThom BWeit Sums
Kitti 243
85
167
1003
185
173
902
14
59
2831
Mad 13
702
582
129
711
48
176
25
158
2544
Sums: 256
787
749
1132
896
221
1078
39
217
5375
From the above raw data we can construct a two way table of results. This type of table is referred to as a pivot table or
cross-tabulation.
Voter Residency
K Kitti M Mad
Candidate residency W Kitti 2321
366
E Mad 510
2178
Sums
2831
2544
Sums
2687
2688
5375
Basic statistical probabilities from the above table
What percentage of voters reside in Kitti?
P(Residency of voter is Kitti K) = P(K) = 2831/5375 = 0.53 = 53%
What percentage of voters reside in Madolehnihmw?
P(Residency of voter is Madolehnihmw M) = P(M) = 2544/5375 = .047 - 47%
- 50 -
What percentage of all votes did Kitti candidates receive?
P(W) = 2687/5375 = .4999 = 49.99%
Try the following at your desk:
What percentage of all votes did Madolehnihmw candidates receive?
P(E) = 2688/5375 = 0.5001 = 50.01%
And
What percentage of the total vote is represented by Kitti residents voting for Kitti candidate?
For AND look at the INTERSECTION and use the number in the intersection.
P(K and W) = 2321/5375 = 0.43 = 43%
Find P(K and E), the percentage of the total vote represented by Kitti residents voting for Madolehnihmw candidates.
P(K and E) = 510/5375 = 0.09 = 9%
Try the following at your desk:
Find P(M and W), the percentage of the total vote represented by Madolehnihmw residents voting for Kitti candidates.
P(M and W) = 366/5375 = 0.07 = 7%
Or
Find P(K or W), the percentage of the total vote represented by all Kitti residents and all voters who voted for a Kitti
candidate. This one is easiest if done by looking at the table. The three cells that have to be added are 2321 + 510 + 366.
This total has to then be divided by the total, 5375.
(2321 + 510 + 366)/5375 = 0.59 = 59%
This can also be calculated from the following formula:
P(A) or P(B) = P(A) + P(B) - P(A or B)
P(K or W) = P(K) + P(W) - P(K and W)
2831/5375 + 2687/5375 - 2321/5375 = 0.5267 + 0.4999 - 0.4318 = 0.59 = 59%
Try the following at your desk:
Find P(K or E), the percentage of the total vote represented by all Kitti residents and all voters who voted for a
Madolehnihmw candidate.
(2321 + 510 + 2178)/5375 = 0.93
Conditional Probability
In conditional probability a specified event has already occurred that affects the remaining statistical probability
calculations. Suppose I want to only look at how the Kitti residents voted, excluding consideration of the
Madolehnihmw voters. I might be asking, "What percentage of Kitti residents (not of the whole vote) voted for Kitti
candidates?" We write this in the following way:
P(W, given K) = 2321/2831 = 0.82 = 82%
Think of the above this way: put your hand over all the Madolehnihmw data and then run your calculations. "K" has
occurred, so we can forget about the "M" column and the sums.
The 82 percent represents, for lack of a better term, a "Kitti loyalty factor." In Kitti, 82 out of 100 hundred residents will
vote for the home municipality candidate, or about 4 out of 5 people.
Try this at your desk:
Find the "Madolehnihmw loyalty factor" P(E, given M):
2178/2544 = 0.86
That is 86 out of 100 residents will vote for the home municipality candidate in Madolehnihmw.
"Cross-over" voting
Find the percentage of Kitti voters who voted "Madolehnihmw" as a percentage of all Kitti voters:
P(E, given K) = 510/2831 = 0.18 = 18%
- 51 -
Call this the "Kitti cross-over factor." 18% of Kitti residents will tend to cross over and vote outside their municipality.
Find the percentage of Madolehnihmw voters who voted "Kitti" as a percentage of all Madolehnihmw voters:
P(W, given M) = 366/2544 = 0.14 = 14%
A campaign statistician for a Kitti candidate might make the following line of reasoning. Only one in seven (~14%)
Madolehnihmw residents is likely to vote Kitti. In some sense, an argument could be made for a Kitti candidate not
spending more than one in seven days campaigning in Madolehnihmw.
On the other hand, one in every five Kitti residents is likely to vote Madolehnihmw. A campaign statistician for a
Madolehnihmw candidate might reasonably recommend spending one in every five days over in Kitti to capitalize on
the cross-over effect.
Another example of dependent events.
Favorite Meat/Favorite Sport
Volleyball
Basketball
Baseball
Hockey
American Football
Pool
Swimming
Sums:
Fish Chicken Dog Sums
FFF F
4
MM M
MM 5
MM
M 3
M
1
F
1
M 1
M
1
12 2
4
18
- 52 -
Probability Distributions
6.1 Types of probabilities and distributions
Mathematically equally likely outcomes usually produce symmetric distributions. Simple probabilities of a single coin
or single die are uniform in their shape. The probabilities of multiple coins or dice form a symmetric heap that is called
a binomial distribution. As the number of dice and pennies increase, the distribution approaches a shape we will later
learn to call the "normal" distribution.
Distributions based on relative frequencies can have a variety of shapes, symmetrical or non-symmetrical.
The shape of the distibution of a sample is often reflective of the shape of the distribution of a population. If the sample
is a good, random sample, then the shape of the sample distribution is a good predictor of the shape of the population
distribution.
Probability Distributions
A probability distribution usually refers to a relative frequency histogram drawn as a line chart.
Both discrete and continuous variables can have a probability distribution. Intervals (or bins or classes) can be
constructed, relative frequencies (or probabilities) can be calculated and a relative frequency histogram can be drawn. If
the data is continuous, then a mean can be calculated for the data from the original data. There is also a way to recover
the mean from the bin values and the probabilities, although this depends on the bin values being treated as being a part
of a continuous distribution. In later chapters the columns of the histogram chart will be replaced by a line, specifically
a "heap" or "mound" shaped line. The diagrams further below show how one might move from a column chart
representation of data to a line chart representation.
The following data consists of 39 body fat measurements for female students at the College of Micronesia-FSM
Summer 2001 and Fall 2001. Following the table is a relative frequency histogram, the probability distribution for this
data.
BFI fem CUL
x
20.1
24.6
29.2
33.7
38.1
Sum (n):
Frequency
f
2
12
13
5
7
39
Relative Frequency
f/n or P(x)
0.05
0.31
0.33
0.13
0.18
1.00
The area under the bars is equal to one, the sum of the relative frequencies. The above diagram consists of five discrete
classes. Later we will look at continuous probability distributions using lines to depict the probability distribution.
Imagine a line connecting the tops of the columns:
- 53 -
If the columns are removed and the class upper limits are shifted to where the right side of each column used to be:
The orange vertical line has been drawn at the value of the mean. This line splits the area under the "curve" in half.
Half of the females have a body fat measurement less than this value, half have a body fat measurement greater than
this value.
We could also draw a vertical line that splits the area under the curve such that we have ten percent of the area to the left
of the orange line and ninety percent to the right of the orange line.
This line would be at the value below which only ten percent of the measurements occur.
6.2 Calculations of the mean and the standard deviation
In some situations we have only the intervals and the frequencies but we do not have the original data. In these
situations it would be useful to still be able to calculate a mean and a standard deviation for our data.
If we only have the intervals and frequencies, then we can calculate both the mean and the standard deviation from the
class upper limits and the relative frequencies. � Here is the mean and standard deviation for the sample of 39 female
students:
BFI fem CUL
x
20.1
24.6
29.2
33.7
38.1
Sum:
Frequency
f
2
12
13
5
7
39
Relative Frequency f/n or
P(x)
0.05
0.31
0.33
0.13
0.18
1.00
Mean μ:
∑(x*P(x))
1.03
7.58
9.72
4.32
6.86
μ = 29.51
A spreadsheet with the above data is available at:
http://www.comfsm.fm/~dleeling/statistics/statistics_fall2001.xls
- 54 -
stdev σ:
√(∑((x-μ)�*P(x)))
4.52
7.29
0.04
2.23
13.56
∑ = 27.64
σ = 5.26
Note that the results are not exactly the same as those attained by analyzing the data directly. Where we can, we will
analyze the original data. This is not always possible. The following table was taken from the 1994 FSM census. Here
the data has already been tallied into intervals, we do not have access to the original data.� Even if we did, it would be
102,724 rows, too many for some of the computers on campus.
Age x Total f
4
9
14
19
24
29
34
39
44
49
54
59
64
69
74
79
84
89
94
98
Sums:
14662
15090
14944
12425
9192
7042
6800
5998
3131
3601
2271
2089
1978
1308
1169
544
313
99
56
12
102724
Relative frequency f/n or
P(x)
0.14
0.15
0.15
0.12
0.09
0.07
0.07
0.06
0.03
0.04
0.02
0.02
0.02
0.01
0.01
0.01
0.00
0.00
0.00
0.00
1
x*P(x) (x-μ)²*P(x)
0.57
1.32
2.04
2.30
2.15
1.99
2.25
2.28
1.34
1.72
1.19
1.20
1.23
0.88
0.84
0.42
0.26
0.09
0.05
0.01
24.12
sqrt:
57.78
33.58
14.90
3.17
0.00
1.63
6.46
12.93
12.05
21.70
19.74
24.74
30.62
25.65
28.31
15.95
10.93
4.06
2.66
0.64
327.50
18.10
The mean μ = 24.12
The population standard deviation σ = 18.10
A spreadsheet with the above data is available from: http://www.comfsm.fm/~dleeling/statistics/statistics.xls
The result is an average age of 24.12 years for a resident of the FSM in 1994 and a standard deviation of 18.10 years.
This means at least half the population of the nation is under 24.12 years old!
Actually, due to the skew in the distribution, fully 56% of the nation is under 19.
Bear in mind that 56% is in school.
That means we will need new jobs for that 56% as they mature and enter the workplace.
On the order of 57,121 new jobs.
How old are you?
Below, at, or above the mean (average)? Do you have a job?
Note we used the class upper limits to calculate the average age. Potentially this inflates the national average by as
much as half a class width or 2.5 years. Taking this into account would yield an average age of 21.62 years old.
There is one more small complication to consider. Since the population of the FSM is growing, the number of people at
each age in years is different across the five year span of the class. The age groups at the bottom of the class (near the
class lower limit) are going to be bigger than the age groups at the top of the class (near the class upper limit). This
would act to further reduce the average age.
Homework: Use the 2000 Census data to calculate the mean age in the FSM in 2000.
Age
4
9
14
19
2000
14782
14168
14213
13230
- 55 -
Age
24
29
34
39
44
49
54
59
64
69
74
79
2000
9527
7620
6480
6016
5560
4650
3205
1903
1733
1487
993
1441
1. Did the mean age change?
2. Are you still (below|at|above) the mean age?
Alternate Homework:
Use the following data to calculate the overall grade point average and standard deviation of the grade point data for the
Pohnpeian students at the national campus during the terms Fall 2000 and Spring 2001
Grade Point Value
x
4
3
2
1
0
Sums:
Frequency
f
851
1120
1023
459
690
______
Relative Frequency
f/n or P(x)
______
______
______
______
______
______
Mean:
∑(x*P(x))
______
______
______
______
______
______
Sqrt:
- 56 -
stdev:
√(∑((x-μ)�*P(x)))
______
______
______
______
______
______
______
Introduction to the Normal Distribution
7.1 Distribution shape
Inferential statistics is all about measuring a sample and then using those values to predict the values for a population.
The measurements of the sample are called statistics, the measurements of the population are called parameters. Some
sample statistics are good predictors of their corresponding population parameter. Other sample statistics are not able to
predict their population parameter.
The sample size will always be smaller than the population. The population size N cannot be predicted from the sample
size n. The sample mode is not usually the same as the population mode. The sample median is also not necessarily a
good predictor of the population median.
The sample mean for a good, random sample, is a good point estimate of the population mean μ. The sample standard
deviation sx predicts the population standard deviation σ. The shape of the distribution of the sample is a good predictor
of the shape of the distribution of the population.
That the shape of the population distribution can be predicted by the shape of the distribution of a good random sample
is important. Later in the course we will be predicting the population mean μ. Instead of predicting a single value we
will predict a range in which the population mean will likely be found.
Consider as an example the following question, "How long does it take to drive from Kolonia to the national campus on
Pohnpei?" A typical answer would be "Ten to twenty minutes." Everyone knows that the time varies, so a range is
quoted. The average time to drive to the national campus is somewhere in that range.
Determining the appropriate range in which a population mean will be found depends on the shape of the distribution. A
bimodal distribution is likely to need a larger range than a symmetrical bell shaped distribution in order to be sure to
capture the population mean.
As a result of the above, we need to understand the shape of distributions generated by different systems. The most
important shape in statistics is the shape of a purely random distribution, like that generated by tossing many pennies.
In class exercise: flipping seven pennies. Student flip seven pennies and record the number of heads. The data for a
section is gathered and tabulated. The students then prepare a relative frequency histogram of the number of heads and
calculate the mean number of heads from Σ x*p(x).
7.2 Seven Pennies
In the table below, seven pennies are tossed eight hundred and fifty eight times. For each toss of the seven pennies, the
number of pennies landing heads up are counted.
# of heads x
7
6
5
4
Frequency
9
112
147
228
Rel Freq P(x)
0.0105
0.1305
0.1713
0.2657
- 57 -
3
2
1
0
195
120
45
2
858
0.2273
0.1399
0.0524
0.0023
1.00
The relative frequency histogram for a large number of pennies is usually a heap-like shape.� For seven pennies the
theoretic shape of an infinite number of tosses can be calculated by considering the whole sample space for seven
pennies
HHHHHHH HHHHHHT HHHHHHTT HHHHTTT HHHTTTT HHTTTTTT HTTTTTT TTTTTTT
HHHHHTH HHHHHTHT HHHTHTT HHTHTTT THTTTTTH TTTTTTH
HHHHTHH HHHHTHHT HHTHHTT HTHHTTT THTTTTHT TTTTTHT
...
...
...
...
...
...
If one works out all the possible combinations then one attains:
(two sides)^(7 pennies) = 128 total possibilities
1 way to get seven heads/128 total possible outcomes = 1/128= 0.0078
7 ways to get six heads and one tail/128 possibilities = 7/128 =0.0547
21 ways to get five heads and two tails/128 = 21/128 = 0.1641
35 ways to get four heads and three tails/128 = 35/128 = 0.2734
35 ways to get three heads and four tails/128 = 35/128 = 0.2734
21 ways to get two heads and five tails/128 = 21/128 = 0.1641
7 ways to get one head and six tails/128 possibilities = 7/128 =0.0547
1 way to get seven tails/128 total possible outcomes = 1/128= 0.0078
If the theoretic relative frequencies (probabilities) are added to our table:
# of heads
x
7
6
5
4
3
2
1
0
Rel Freq
Theoretic
P(x)
9
0.0105 0.0078
112
0.1305 0.0547
147
0.1713 0.1641
228
0.2657 0.2734
195
0.2273 0.2734
120
0.1399 0.1641
45
0.0524 0.0547
2
0.0023 0.0078
858
1.00
1.00
If the theoretic relative frequencies are added as a line to our graph, the following graph results:
Frequency
- 58 -
The gray line represents the shape of the distribution for an infinite number of coin tosses. The shape of the distribution
is
symmetrical.
If both the number of pennies is increased as well as the number of tosses, then the graph would become smoother and
increasingly symmetrical. Below is a graph for tens of thousands of tosses of 21 pennies.
7.3 The Normal Curve
If the number of pennies and tosses are both allowed to go to infinity, then a smooth curve results looking a lot like the
curve seen above. The smooth curve that results can be described by a function. Statistical mathematicians would say
that as the number of sides and tosses approaches infinity, the discrete distribution approaches a continuous distribution
described by the function below.
In the above function, σ is the population standard deviation, μ is the population mean, e is the base e, and π is pi. The
name of this function is the "normal" curve. I like to think of it as being called normal because it is what "normally"
happens if you toss a lot of pennies a lot of times! If the above function is graphed for a mean μ = 0 and a population
standard deviation σ = 1, then the following graph results:
- 59 -
The above function has the following properties:
•
•
•
•
•
•
•
•
symmetrical about μ = 0
"bell" shaped
highest probability at μ = 0
approaches x-axis but never crosses (asymptotic to the x-axis)
the numbers on the x-axis are the number of standard deviations away from the mean
transition (inflection) points at μ ± 1σ
the area under any portion of the curve is the probability of x being within that span
the area under the curve between μ - σ and μ + σ is 0.6826, thus the probability that an x value is between μ - σ
and μ + σ is 68.26%
The area under each "section" of the normal curve can be seen in the following diagram.
- 60 -
For example, the area under the curve beyond (to the right of) μ + 2σ is 0.0228 or 2.28%. The probability of a data
value being greater than μ + 2σ is 0.0228. A data value could be expected out here once in about 44 instances.
6σ: "Six sigma" A business quality program that attempts to bring error down to 3 in a million (μ + 6σ)
The shape of the normal curve is affected by the standard deviation. In the diagram below m is the mean μ and sx is the
standard deviation.
Changes to the mean shift the normal curve horizontally:
- 61 -
How relative frequencies become area under a curve
Let us begin with a more familiar example from our work earlier in the term.
Heap like shapes often result from histograms of data. The following is a frequency table for the height data for 60
females in statistics class in an earlier term.
Female height CUL Frequency Relative Frequency
59.6
6
0.10
61.2
16
0.27
62.8
18
0.30
64.4
16
0.27
66
4
0.07
Sums: 60
1.00
The following relative frequency histogram for the heights of 60 females above has the following distribution:
Imagine changing this discrete distribution into a continuous distribution.
- 62 -
The probability distribution above says that 10% of the women are less than or equal to 59.6 inches tall. 27% of the
women measured are taller than 59.6 inches and shorter than or equal to 61.2 inches. What is the probability of finding
a female student taller than 64.4 inches tall? Seven percent. The area "under" each segment of the "curve" is the
probability of a women being in that range of heights.
The difficulty with the above analysis is seen in attempting to answer the following question: What percentage of
female students are taller than 60 inches? This cannot easily be determined from the above data. An answer could be
interpolated, but that would be the best we would be able to do.
In some instances the actual shape of the population distribution is not exactly known, but the distribution is expected to
be heaped, to behave "normally" and heap up in the manner of the normal distribution.
Because there is a mathematical equation for the normal distribution, the probabilities (the areas under the curve!) can
be determined mathematically.
A Normal Curve Example
Suppose we know that sixty customers arrive at a sakau market on a Friday night at a mean time of μ = 7:00 P.M. with a
standard deviation of σ = 30 minutes (0.5 hours). Suppose also that the time of arrival for the customers is normally
distributed (note that areas are rounded).
- 63 -
•
•
•
•
We would expect 0.50 of the customers to arrive by 7:00. 7:00 is the mean value, the middle of the normal
curve, half-way. That would be equal to: 60 * 0.50 = 30 customers by 7:00.
We would expect 0.341 or 34.1% of the customers to arrive between 6:30 and 7:00. That would be 60 * 0.341
= 20.46 or about 21 customers.
0.682 or 68.2% of the customers should arrive between 6:30 ( -1 σ) and 7:30 (+1 σ). Here is the origin of of my
saying that the "68%" of the students have performed between μ - σ and μ + σ on a test if the test scores are
normally distributed.
Note that we cannot do calculations such as, "How many customers have arrived by 6:45?" because our graph
does not include 6:45. We can only make calculations on integer numbers of standard deviations away from the
mean.
Note that in the above example the population mean μ and population standard deviation σ are used. Our normal
distribution work is based on a theories that use the population parameters. Later in the course we will use a modified
normal distribution called the student's t-distribution to work with sample statistics such as the sample mean x and the
sample standard deviation sx for small samples. For many examples in this text, the population parameters are not
known. Until the student's t-distribution is introduced, data that forms a reasonably "heap-like" shape will be analyzed
using the normal distribution.
7.4 The area to the left of x
x to probability p
Areas to the left of x
The probability p is the same as the area under the normal curve. Probability, expressed often as a percentage, is area.
Probability is also the relative frequency. In this class probability, p, area, and relative frequency are all used
interchangeably.
If x is not an whole number of standard deviations from the mean, then we cannot use a diagram as seen above.
Spreadsheets have a function that calculates the area (probability) to the left of ANY x value. The letter p for probability
is used for the area to the left of x.
The function that calculates the area to the left of x is:
=normdist(x;μ;σ;1)
The mean height μ for 43 female students in statistics is 62.0 inches with a standard deviation of 1.9. Determine the
probability that a student is less than 60 inches tall (five feet tall).
In OpenOffice the probability p =
=normdist(60;62;1.9;1)
=0.1463
14.63% of the area is to the left of 60 inches. The probability a female student in statistics class is below 60 inches is
14.63%.
Notation note: In probability notation the above would be written p(x < 60) = 0.1463
When the words "less than, smaller, shorter, fewer, up to and including" are used then the NORMDIST function can be
used to calculate the probability.
- 64 -
Area to the right of x
The mean number of cups of sakau consumed in sakau markets on Pohnpei is μ = 3.65 with a standard deviation of σ =
2.52. Note that this data is actually based on customer data for 227 customers at four markets - one near Kolonia and
three in Kitti. Although this data is actually sample data and not population data, we will treat the mean and standard
deviation as population parameters. The data is not perfectly normally distributed. The data is, however, distributed in a
reasonably smooth heap.
What is the probability a customer will drink more than five cups?
Note the word "more." If the question were "What is the probability that a customer will drink less than five cups, then
the solution would be =NORMDIST(5;3.65;2.52;1). This result is 0.70 or a 70% probability a customer will drink less
than five cups.
The area under the whole normal curve is 1.00. Remember that 1.00 is also 100%. If 70% drink less than five cups, then
we can calculate the probability that those who drink more than five cups is 30%. 100% − 70% = 30%.
Or 1.00 − 0.70 = 0.30
Making a sketch of the normal curve including the mean, the x-value, and the area of interest can help determine when
to subtract a result from one and when to not.
Area between two x values
A study of the prevalence of diabetes in a village on Pohnpei found a mean fasting blood sugar level of μ = 117 with a
standard deviation σ = 33 in mg/dl for females aged 20 to 29 years old. Blood sugar levels between 120 and 130 are
considered borderline diabetes cases. What percentage of the females aged 20 to 29 years old in this village are between
a mean fasting blood sugar of 120 and 130 mg/dl?
For this example, presume that the distribution is normal.
The probability is the percentage. The probability is the area between x = 120 and x = 130 as seen in the image below.
In probability notation this would be written p(120 < x < 130) = ?
To obtain the area between 120 and 130, calculate the area to the left of 120.
- 65 -
Then calculate the area to the left of 130.
Subtract the area to the left of 120 from the area to the left of 130. What remains is the area between 120 and 130.
The table below represents a spreadsheet laid out to calculate the area to the left of 120 in column B and the area to the
left of 130 in column C.
A
B
C
D
1 x
120
130
2 mean μ 117
117
3 stdev σ 33
33
4 normdist =NORMDIST(B1;B2;B3;1) =NORMDIST(C1;C2;C3;1) =C4-B4
4 normdist 0.54
0.65
0.11
Row four is presented twice: once with the formulas and once with the results of the formulas.
The area to the left of 120 is 0.54. The area to the left of 130 is 0.65. 0.65 − 0.54 is 0.11. The probability that females
aged 20 to 29 years old in this village have a blood sugar level between 120 and 130 is 11%.
7.5 Area to x
Conversely, given a probability, a mean, and a standard deviation, an x value can be calculated.
On the college essay admissions test a perfect score is 40. In a recent spring run of the admissions test the mean score
was 21 and the standard deviation was 12. Below what score x are the lowest 33% of the student scores? Presume that
the data is normally distributed.
In this case we have an area. Percentages are probabilities. Probabilities are area under the curve. We do not know x. To
- 66 -
find the area to the left of x the function NORMINV is used. The letter p is the probability, the area.
=NORMINV(p;μ;σ)
In this case area =NORMINV(0.33;21;12). Note that the area is expressed as a decimal. Alternatively the area could
be entered as 33%. The result of this calculation is 15.72. 33% of the students scored below a 15.72 on the essay test.
Area to the right of x
Suppose the height of women at the College is normally distributed with a mean of 62.0 inches and a standard deviation
of 1.9 inches. Suppose I want to know the minimum height of the top 10% of the female students at the College.
In this instance I have a probability, the top 10%. The NORMINV function, however, requires the area to the right of x.
If the area to the right is 10%, then the area to the left is 100% − 10% = 90%.
area =NORMINV(0.90;62;1.9)
The result is 64.43.
Thus the minimum height of the top 10% is 64.43 inches. If there are 350 women at the college, then 0.10 * 350 = 35
women can be expected to be taller than 64.4 inches.
Domino's pizza knows that the average length of time from receiving an order to delivering to the customer is 20
minutes with a standard deviation 7 min 45 seconds. Treat these sample statistics as population parameters for now.
Dominoes wants to guarantee a delivery time as part of a marketing campaign, "Your pizza in ___ minutes of your
money back!" Dominoes is willing to refund 10% of their orders, what is the quickest delivery time they should set the
grantee at?
The area to the left of x is 90% therefore the correct function is =NORMINV(0.9;20;7.75)
The result is 29.92 minutes
So you guarantee delivery in 30 minutes or less and you'll only pay out on 10% of the pizzas. (From another perspective
this is a "Buy ten to get one free program").
- 67 -
Sampling Distribution of the Mean
8.1 Distribution of Statistics
As noted in earlier chapters, statistics are the measures of a sample. The measures are used to characterize the sample
and to infer measures of the population termed parameters.
Parameter
A parameter is a numerical description of a population. Examples include the population mean � and the population
standard deviation σ.
Statistic
A statistic is a numerical description of a sample. Examples include a sample mean x and the sample standard deviation
sx.
Good samples are random samples where any member of the population is equally likely to be selected and any sample
of any size n is equally likely to be selected. Consider four samples selected from a population. The samples need not be
mutually exclusive as shown, they may include elements of other samples.
The sample means x1, x2, x3, x4, can include a smallest sample mean and a largest sample mean. Choosing a number of
bins can generate a histogram for the sample means. The question this chapter answers is whether the shape of the
distribution of sample means from a population is any shape or a specific shape.
Sampling Distribution of the Mean
The shape of the distribution of the sample mean is not any possible shape. The shape of the distribution of the sample
mean, at least for good random samples with a sample size larger than 30, is a normal distribution. That is, if you take
random samples of 30 or more elements from a population, calculate the sample mean, and then create a relative
frequency distribution for the means, the resulting distribution will be normal.
In the following diagram the underlying data is bimodal and is depicted by the light blue columns. Thirty data elements
were sampled forty times and forty sample means were calculated. A relative frequency histogram of the sample means
is plotted in a heavy black outline. Note that though the underlying distribution is bimodal, the distribution of the forty
means is heaped and close to symmetrical. The distribution of the forty sample means is normal.
In the following diagram the underlying data is bimodal and is depicted by the columns with thin outlines. Thirty data
elements were sampled forty times and forty sample means were calculated. A relative frequency histogram of the
sample means is plotted with a heavy black outline. Note that though the underlying distribution is bimodal, the
distribution of the forty means is heaped and close to symmetrical. The distribution of the forty sample means is
- 68 -
normal.
The center of the distribution of the sample means is, theoretically, the population mean. To put this another simpler
way, the average of the sample averages is the population mean. Actually, the average of the sample averages
approaches the population mean as the number of sample averages approaches infinity.
Another Example (2002)
Consider a population consisting of 61 body fat measurements for women at the COM-FSM national campus:
15.6, 18.9, 20, 20.3, 20.6, 20.8, 21.9, 22.1, 22.2, 22.2, 22.4, 22.7, 22.8, 22.8, 23.5, 23.5, 23.6, 23.8, 23.9, 24.3, 24.4,
25.2, 25.2, 25.5, 25.6, 26.1, 26.2, 27.3, 27.5, 27.8, 27.9, 28, 28, 28.1, 28.1, 28.3, 28.4, 29.2, 29.3, 29.3, 29.5, 29.8, 30.5,
31.1, 31.6, 32.9, 34, 34.4, 34.9, 35.5, 35.8, 35.9, 36, 37.5, 38.2, 38.8, 40, 40.8, 44.1, 47, 50.1
The population mean (parameter)for the above data is 28.7. Consider those measurements as being the total population.
The distribution of those measurements using an eight bin histogram is seen below.
Bin
19.9
24.2
28.5
32.9
37.2
41.5
45.8
50.1
Freq
2
17
18
8
8
5
1
2
61
RelFreq
0.03
0.28
0.30
0.13
0.13
0.08
0.02
0.03
1.00
The distribution is skewed right, as seen above.
- 69 -
If we were doing a statistical study, we would measure a random sample of women from the population and calculate
the mean body fat for our sample. Then we would use our sample statistic (our sample mean) to estimate the population
parameter (the population mean). Understanding the SHAPE of the distribution of many sample means is a key to using
a single sample mean (a statistic) to estimate the population mean (a parameter).
The table that follows consists of ten randomly selected samples from the population and the means for each sample.
Each sample has a size of n=10 women. The bottom row is the mean of each sample.
Smpl 1 Smpl 2 Smpl 3 Smpl 4 Smpl 5 Smpl 6 Smpl 7 Smpl 8 Smpl 9 Smpl 10
40.8
40
20.3
24.3
21.9
44.1
22.8
22.1
34.4
50.1
40.8
38.2
27.3
25.2
28.3
38.2
20
29.5
20.8
29.2
34
27.5
28
35.9
27.9
29.2
38.8
25.6
31.6
35.5
26.1
35.5
40
23.9
23.8
22.8
24.4
22.2
38.2
28.3
20.3
27.5
34.9
27.8
32.9
20.6
29.8
27.3
28.1
22.8
25.2
32.9
34
23.6
29.3
25.6
38.2
27.8
20.3
20.3
30.5
25.6
29.3
35.5
22.4
27.8
26.2
30.5
22.7
24.4
37.5
40
23.9
29.5
28.4
24.4
29.2
36
31.1
36
40
34.4
28
23.6
27.8
31.1
25.2
20.8
47
34
15.6
27.3
20.8
31.6
35.8
28
35.8
31.1
22.2
22.4
31.08 32.89 28.65 28.09 27.85 29.18 29.04 27.29 29.64
30.3
The mean of the values in the last row is 29.4. This could be called the mean of the sample means! A histogram can be
used to show the distribution of these sample means. These frequencies and relative frequencies are in the two rightmost
columns of the table below.
Bin
19.9
24.2
28.5
32.9
37.2
41.5
45.8
50.1
Freq RelFreq AvgDist RFavg
2
0.03
0
0
17
0.28
0
0
18
0.30
3
0.3
8
0.13
6
0.6
8
0.13
1
0.1
5
0.08
0
0.0
1
0.02
0
0.0
2
0.03
0
0.0
61
1.00
10
1.00
Note that the sample means are clustered tightly about the population mean. This can be seen below where the sample
mean distribution is superimposed (placed on top of!) the population distribution.
The Shape of the Sample Mean Distribution is Normal!
The sample mean distribution is a heap shaped, as in the shape of the normal distribution, and centered on the
population mean.
If the sample size is 30 or more, then the sample means are NORMALLY distributed even when the underlying data is
NOT normally distributed! If the sample size is less than 30, then the distribution of the samples means is normal if and
only if the underlying data is normally distributed.
- 70 -
The normal distribution of the sample means (averages) allows us to use our normal distribution probabilities to
estimate a range for μ. The mean of the sample means is a point estimate for the population mean μ.
The mean of the sample means can be written as:
μ
The value of the mean of the sample means μ is, for a very large number of samples each of which has a very large
In this text the above is sometimes written as
x
x
sample size, the population mean. As a practical matter we use the mean of a single large sample. How large? The
sample size must be at least n = 30 in order for the sample mean (a statistic) to be a good estimate for the population
mean (a parameter). This requirement is necessary to ensure that the distribution of the sample means will be normal
even when the underlying data is not normal. If we are certain the data is normally distributed, then a sample size n of
less than 30 is acceptable.
Later in the course we will modify the normal distribution to handle samples of sizes less than 30 for which the
distribution of the underlying data is either unknown or not normal. This modification will be called the student's tdistribution. The student's t-distribution is also heap-shaped.
The normal distribution, and later the student's t-distribution, will be used to quote a range of possible values for a
population mean based on a single sample mean. Knowing that the sample mean comes from a heap-shaped distribution
of all possible means, we will center the normal distribution at the sample mean and then use the area under the curve to
estimate the probability (confidence) that we have "captured" the population mean in that range.
8.2 Central Limit Theorem
The Central Limit Theorem is the theory that says "for increasingly large sample sizes n, the sample mean approaches
ever closer the population mean."
Standard Error
The standard deviation of the distribution of the sample means
There is one complication: the sample standard deviation of a single sample is not a good estimate of the standard
deviation of the sample means. Note that the distribution of the sample means is NARROWER than the sample in the
above example. The shape of the distribution of the sample means is narrower and taller than the shape of the
underlying data. In the diagram, the shape of the underlying data is normal, the taller narrower distribution is the
distribution of all the sample means for all possible samples.
The standard deviation of a single sample has to be reduced to reflect this. This reduction turns out to be inversely
related to the square root of the sample size. This is not proven here in this text.
The standard deviation of the distribution of the sample means is equal to the actual population standard deviation
divided by the square root of n.
The standard deviation divided by the square root of the sample size is called the standard error of the mean.
- 71 -
If σ is known, then the above formula can be used and the distribution of the sample mean is normal.
As a practical matter, since we rarely know the population standard deviation σ, we will use the sample standard
deviation sx in class to estimate the standard deviation of the sample means. This formula will then appear in various
permutations in formulas used to estimate a population mean from a sample mean. When we use the sample standard
deviation sx we will use the student's t-distribution. The student's t-distribution looks like a normal distribution. The
student's t-distribution, however, is adjusted to be a more accurate predictor of the range for a population mean. Later
we will learn to use the student's t-distribution. Until that time we will play a little fast and loose and use sample
standard deviations to calculate the standard error of the mean.
- 72 -
Confidence Intervals
9.1 Inference and Point Estimates
Whenever we use a single statistic to estimate a parameter we refer to the estimate as a point estimate for that
parameter. When we use a statistic to estimate a parameter, the verb used is "to infer." We infer the population parameter
from the sample statistic.
Some population parameters cannot be inferred from the statistic. The population size N cannot be inferred from the
sample size n. The population minimum, maximum, and range cannot be inferred from the sample minimum,
maximum, and range. Populations are more likely to have single outliers than a smaller random sample.
The population mode and median usually cannot be inferred from a smaller random sample. There are special
circumstances under which a sample mode and median might be a good estimate of a population mode and median,
these circumstances are not covered in this class.
The statistic we will focus on is the sample mean x. The normal distribution of sample means for many samples taken
from a population provides a mathematical way to calculate a range in which we expect to "capture" the population
mean and to state the level of confidence we have in that range's ability to capture the population mean μ.
Point Estimate for the population mean μ and Error
The sample mean x is a point estimate for the population mean μ
The sample mean x for a random sample will not be the exact same value as the true population mean μ.
The error of a point estimate is the magnitude of estimate minus the actual parameter (where the magnitude is always
positive). The error in using x for μ is ( x − μ ). Note that to take a positive value we need to use either the absolute
value |( x - μ )| or √( x - μ )2.
Note that the error of an estimate is the distance of the statistic from the parameter.
Unfortunately, the whole reason we were using the sample mean x to estimate the population mean μ is because we did
not know the population mean μ.
For example, given the mean body fat index (BFI) of 51 male students at the national campus is x = 19.9 with a sample
standard deviation of sx = 7.7, what is the error |( x - μ )| if μ is the average BFI for male COMFSM students?
We cannot calculate this. We do not know μ! So we say x is a point estimate for μ. That would make the error equal to
√(x − x)2 = zero. This is a silly and meaningless answer.
Is x really the exact value of μ for all the males at the national campus? No, the sample mean is not going to be the same
as the true population mean.
Point estimate for the population standard deviation σ
The sample standard deviation sx is a reasonable point estimate for the population standard deviation σ. In more
advanced statistics classes concern over bias in the sample standard deviation as an estimator for the population
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standard deviation is considered more carefully. In this class, and in many applications of statistics, the sample standard
deviation sx is used as the point estimate for the population standard deviation σ.
9.12 Introduction to Confidence Intervals
We might be more accurate if we were to say that the mean μ is somewhere between two values. We could estimate a
range for the population mean μ by going one standard error below the sample mean and one standard error above the
sample mean. Remember, the standard error is the σ/√(n). Note that the formula for the standard error requires knowing
the population standard deviation σ. We do not usually know this value. In fact, if we knew σ then we would probably
also know the population mean μ. In section 9.2 we will use the sample standard deviation or sx/√(n) and the student's tdistribution to calculate a range in which we expect to find the population mean μ.
In the diagram the lower curve represents the distribution of data in a population with a normal distribution. Remember,
distribution simply means the shape of the frequency or relative frequency histogram, now charted as a continuous line.
The narrower and taller line is the distribution of all possible sample means from that population.
For the population curve (lower, broader) the distance to each inflection point is one standard deviation: ± σ. For the
sample means (higher, narrower) the distance to each inflection point is one standard error of the mean: ± σ/√(n).
The area from minus one standard error to plus one standard error is still 68.2%.
Here is a key point: If I set my estimate for μ to be between x - σ/√n and x + σ/√n, then there is a 0.682 probability that
μ will be included in that interval.
The "68.2% probability" is termed "the level of confidence."
Probability note: the reality is that the population mean is either inside or outside the range we have calculated. We are
right or wrong, 100% or 0%. Thus saying that there is a 68.2% probability that the population mean has been "captured"
by the range is not actually correct. This is the main reason why we shift to calling the range for the mean a confidence
interval. We start saying things such as "I am 68.2 percent confident the mean is in the range quoted." Statisticians
assert that over the course of a lifetime, if one always uses a 68.2% confidence interval one will right 68.2% of the time
in life. This is small comfort when an individual experimental result might be very important to you.
95% Confidence Intervals
In many fields of inquiry a common level of confidence used is a 95% level of confidence. For the purposes of this
course a 95% confidence interval is often used.
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9.18 Confidence Intervals for n > 30 where σ is known
The sampling distribution of the mean is a normal distribution with the standard error replacing the standard deviation.
The diagram above shows the 95% area under the curve. The NORMINV function can find the left and right values for
the range in which we expect the mean to be found 95% of the time. This range is called the 95% confidence interval. In
the diagram the ends of the range are indicated by the lower and upper limits.
=NORMINV(p;μ;σ/√(n))
The NORMINV function uses the area to the left of the lower limit to find the lower limit. That area can be determined
by noting that the whole area under the curve is 100%. This means that 5% is distributed in two equal tails. Each tail is
half of 5%. Each tail is 2.5 or 0.025 in decimal notation. Thus the lower limit can be found by using the area 0.025.
The upper limit can be found by using the area to left of the upper limit. The area to the left of the upper limit is 2.5% +
95%. This is 97.5% or 0.975 in decimal notation.
Example
Find the 95% confidence interval for the population mean number of cups of sakau en Pohnpei consumed by a
customer. The sample consists of 227 customers who drank an average 3.65 cups of sakau with a standard deviation of
2.52.
While we lack the population standard deviation, the sample is large enough and the underlying data is sufficiently
heap-like that the sample standard deviation is a good point estimate for the population standard deviation. In this
example n = 227, x = 3.65; and sx = 2.52. Note that x and sx are being used to estimate μ and σ
The lower ("left") limit for the population mean:
=NORMINV(0.025;3.65;2.52/SQRT(227))
The result is 3.32 cups.
The upper ("right") limit for the population mean:
=NORMINV(0.975;3.65;2.52/SQRT(227))
The result is 3.98 cups.
Remember: the p in the NORMINV function is the area to the left of the x-axis value. For 95% of the area under the
curve, the amount of area in the "tails" is 5%. Half in the left, half in the right. The right tail is 2.5% or 0.025. The left
tail is also 2.5%, but the area to the LEFT of this 2.5% is 97.5% or 0.975.
Margin of Error E of the mean
The Margin of Error E for the mean is the distance from the sample mean x to either one of the ends of the confidence
interval. The margin of error E is always calculated to come out positive. For the example above:
=3.65 − 3.32
=3.98 − 3.65
The margin of error E is 0.33. This represents an uncertainty at a 95% level of confidence of one third of a cup of sakau.
The confidence interval is often written as:
x-E≤μ≤x+E
For the sakau cup study the 95% confidence interval would be written 3.32 ≤ μ ≤ 3.98.
Another common notation you will sometimes see is to write the sample mean x ± margin of error. For the example
above we could write: 3.65 ± 0.33
A third notation is related to probability notation: p(3.32 ≤ μ ≤ 3.98) = 0.95 This is related to the first format above and
is rarely seen in publications.
Standard of Error of the mean, Margin of Error for the mean
Do not confuse these two terms. The Standard Error of the mean is ± σ/√(n). The Margin of Error for the mean is the
distance from either end of the condifendence interval to the middle of the confidence interval.
Example:
Given that n = 219 CHS students took the TOEFL examination with a sample mean score of x = 369 and a sample
standard deviation sx = 50, construct a 90% confidence interval for the population mean TOEFL score for CHS.
The point estimate for the population mean μ is 369.
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To find the lower limit use the area to the left of the lower limit. The area in the "left tail" can be found by noting that
both tails must be 100% − 90% = 10%. So each "tail" has an area that is half of 10%. The left tail is 5%.
=NORMINV(0.05;369;50/SQRT(219))
The result is 363.44 for the lower limit of the 90% confidence interval for the population mean μ.
The upper limit is found from the area to the left of the upper limit. Use a sketch to help determine this area. The area to
the left is the 5% in the left tail plus the 90% confidence area or 95%
The confidence interval will be given by:
=NORMINV(0.95;369;50/SQRT(219))
The result is 374.56 for the upper limit of the 90% confidence interval for the population mean μ.
Using the notation x - E ≤ μ ≤ x + E we would write:
363.44 ≤ μ ≤ 374.56
When speaking we would say that we have a 90% level of confidence in having captured the population mean. The
level of confidence is sometimes noted by the letter c.
In the next section we will tackle what happens when the sample size n is less than thirty. The function we will use will
also work for sample sizes larger than 30. The function, however, will calculate the margin of error E, not the lower and
upper limits. For the above example the margin of error E could be calculated from either limit. Either take the sample
mean minus the lower limit or the upper limit minus the sample mean:
=374.56 − 369
=369 − 363.44
Either way the result is 5.56. The margin of error E is 5.56. Another way to write this 90% confidence interval would
be:
90% CI: 369 ± 5.56
Note that when a confidence interval is not 95%, then specific reference to the chosen confidence level must be stated.
Stating the level of confidence is always good form. While many studies are done at a 95% level of confidence, in some
fields higher or lower levels of confidence may be common. Scientific studies often use 99% or higher levels of
confidence.
There is always, however, a chance that one will be wrong. In Florida an election was "called" in favor of candidate Al
Gore in the year 2000 in the United States based on a 99.5% level of confidence. Hours later the news organizations
said George Bush had won Florida. A few hours later the news organizations would retract this second estimate and
decide that the race was too close too call. The news organizations decided they had been wrong two times in row.
Eventually a court case finally settled who had won the state of Florida. Even at a half a percent chance of being wrong
one can still be wrong, even two times in a row.
Confidence intervals for n ≥ 5 using sx
9.2
When using the sample standard deviation sx to generate a confidence interval for the population mean, a distribution
called the Student's t-distribution is used. The Student's t-distribution looks like the normal distribution, but the tdistribution changes shape slightly as the sample size n changes. The t-distribution looks like a normal distribution, but
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the shape "flattens" as n decreases. As the sample size decreases, the t-distribution becomes flatter and wider, spreading
out the confidence interval and "pushing" the lower and upper limits away from the center. For n ≥ 30 the Student's tdistribution is almost identical to the normal distribution. When we sketch the Student's t-distribution we draw the same
heap shape with two inflection points.
To use the Student's t-distribution the sample must be a good, random sample. The sample size can be as small as n = 5.
For n ≤ 10 the t-distribution will generate very large ranges for the population mean. The range can be so large that the
estimate is without useful meaning. A basic rule in statistics is "the bigger the sample size, the better."
The spreadsheet function used to find limits from the Student's t-distribution does not calculate the lower and upper
limits directly. The function calculates a value called "t-critical" which is written as tc. t-critical muliplied by the
Standard Error of the mean SE will generate the margin of error for the mean E.
Do not confuse the standard error of the mean with the margin of error for the mean. The Standard Error of the mean is
sx/√(n). The Margin of Error for the mean (E) is the distance from either end of the condifendence interval to the middle
of the confidence interval. The margin of error is produced from the Standard Error:
Margin of Error for the mean = tc*standard error of the mean
Margin of Error for the mean = tc*sx/√n
The confidence interval will be:
x-E≤μ≤x+E
Calculating tc
The t-critical valus will be calculated using the spreadsheet function TINV. TINV uses the area in the tails to calculate tcritical. The area under the whole curve is 100%, so the area in the tails is 100% − confidence level c. Remember that in
decimal notation 100% is just 1. If the confidence level c is in decimal form use the spreadsheet function below to
calculate tc:
=TINV(1−c;n−1)
If the confidence level c is entered as a percentage with the percent sign, then make sure the 1 is written as 100%:
=TINV(100%−c%;n−1)
Degrees of Freedom
The TINV function adjusts t-critical for the sample size n. The formula uses n − 1. This n − 1 is termed the "degrees of
freedom." For confidence intervals of one variable the degrees of freedom are n − 1.
Example 9.2.1
Runners run at a very regular and consistent pace. As a result, over a fixed distance a runner should be able to repeat
their time consistently. While individual times over a given distance will vary slightly, the long term average should
remain approximately the same. The average should remain within the 95% confidence interval.
For a sample size of n = 10 runs from the college in Palikir to Kolonia town, a runner has a sample mean x time of 61
minutes with a sample standard deviation sx of 7 minutes. Construct a 95% confidence interval for my population mean
run time.
Step 1: Determine the basic sample statistics
sample size n = 10
sample mean x = 61
[61 is also the point est. for the pop. mean μ]
sample standard deviation sx = 7
Step 2: Calculate degrees of freedom, tc, standard error SE
degrees of freedom = 10 - 1 = 9
tc =TINV(1-0.95,10-1) = 2.2622
Standard Error of the mean sx/√n = 7/sqrt(10) = 2.2136
Keeping four decimal places in intermediate calculations can help reduce rounding errors in calculations. Alternatively
use a spreadsheet and cell references for all calculations.
Step 3: Determine margin of error E
Margin of error E for the mean
= tc*sx/√n = (2.4469)*(7)/√10 = 5.01
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Given that: x - E ≤ μ ≤ x + E, we can substitute the values for x and E to obtain the 95% confidence interval for the
population mean μ:
Step 4: Calcuate the confidence interval for the mean
61 − 5.01 ≤ μ ≤ 61 + 5.01
55.99 ≤ μ ≤ 66.01
I can be 95% confident that my population mean μ run time should be between 56 and 66 minutes.
Example 9.2.2
Jumps
102 66 42 22 24 107 8 26 111
79 61 45 43 10 17 20 45 105
68 69 79 13 11 34 58 40 213
On Thursday 08 November 2007 a jump rope contest was held at a local elementary school festival. Contestants jumped
with their feet together, a double-foot jump. The data seen in the table is the number of jumps for twenty-seven female
jumpers. Calculate a 95% confidence interval for the population mean number of jumps.
The sample mean x for the data is 56.22 with a sample standard deviation of 44.65. The sample size n is 27. You should
try to make these calculations yourself. With those three numbers we can proceed to calculate the 95% confidence
interval for the population mean μ:
Step 1: Determine the basic sample statistics
sample size n = 27
sample mean x = 56.22
sample standard deviation sx = 44.65
Step 2: Calculate degrees of freedom, tc, standard error SE
The degrees of freedom are n − 1 = 26
Therefore tcritical = TINV(1-0.95;27-1) = 2.0555
The Standard Error of the mean SE = sx/√27 = 8.5924
Step 3: Determine margin of error E
Therefore the Margin of error for the mean E
tc* SE = 2.0555*8.5924 = 17.66
The 95% confidence interval for the mean is x − E ≤ μ ≤ x + E
Step 4: Calcuate the confidence interval for the mean
56.22 − 17.66 ≤ μ ≤ 56.22 + 17.66
38.56 ≤ μ ≤ 73.88
The population mean for the jump rope jumpers is estimated to be between 38.56 and 73.88 jumps.
9.3 Confidence intervals for a proportion
In 2003 a staffer at the Marshall Islands department of education noted in a newspaper article that Marshall's Island
public school system was not the weakest in Micronesia. The staffer noted that Marshall's was second weakest,
commenting that education metrics in the Marshall's outperform those in Chuuk's public schools.
In 2004 fifty students at Marshall Islands High School took the entrance test. Ten students Achieved admission to
regular college programs. In Chuuk state 7% of the public high school students gain admission to the regular college
programs. If the 95% confidence interval for the Marshall Islands proportion includes 7%, then the Marshallese students
are not academically more capable than the Chuukese students, not statistically significantly so. If the 95% confidence
interval does not include 7%, then the Marshallese students are statistically significantly stronger in their admissions
rate.
ˆFinding the 95% confidence interval for a proportion involves estimating the population proportion p. The fifty
students at Marshall Islands High School are taked as a sample. The proportion who gained admission, 10/50 or 20%, is
the sample proportion. The population proportion is treated as unknown, and the sample proportion is used as the point
estimate for the population proportion.
Note: In this text the letter p is used for the sample proportion of successes instead of "p hat". A capital P is used to refer
to the population proportion.
The letter n refers to the sample size. The letter p is the sample proportion of successes. The letter q is the sample
proportion of failures. In the above example n is 50, p is 10/50 or 0.20, and q is 40/50 or 0.80
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Estimating the population proportion P can only be done if the following conditions are met:
np > 5
nq > 5
In the example np = (50)(0.20) = 10 which is > 5. nq = (50)(0.80) = 40 which is also > 5
The standard error of a proportion is √(pq/n). For the example above the standard error is:
=sqrt((0.2)*(0.8)/50)
The standard error for the proportion is 0.0566. The margin of error E is then calculated in much the same way as in the
section above, by multiplying tc by the standard error. tc is still found from the TINV function. The degrees of freedom
will remain n-1.
The margin of error E = tc*√(pq/n)
=TINV(1-0.95;50-1)*sqrt((0.2)*(0.8)/50)
The margin of error E is 0.1137
The confidence interval for the population proportion P is:
p -E ≤ P≤ p +E
0.8 − 0.1137 ≤ P ≤ 0.8 + 0.1137
0.0863 ≤ P ≤ 0.3137
The result is that the expected population mean for Marshall Island High School is between 8.6% and 31.2%. The 95%
confidence interval does not include the 7% rate of the Chuuk public high schools. While the college entrance test is not
a measure of overall academic capability, there are few common measures that can be used across the two nations. The
result does not contradict the staffer's assertion that MIHS outperformed the Chuuk public high schools. This lack of
contradiction acts as support for the original statement that MIHS outperformed the public high schools of Chuuk in
2004.
Homework: In twelve sumo matches Hakuho bested Tochiazuma seven times. What is the 90% confidence interval for
the population proportion of wins by Hakuho over Tochiazuma. Does the interval extend below 50%? A commentator
noted that Tochiazuma is not evenly matched. If the interval includes 50%, however, then we cannot rule out the
possibility that the two-win margin is random and that the rikishi (wrestlers) are indeed evenly matched.
Hakuho won that night, upping the ratio to 8 wins to 5 losses to Tochiazuma. Is Hakuho now statistically more likely to
win or could they still be evenly matched at a confidence level of 90%?
9.4 Deciding on a sample size
Suppose you are designing a study and you have in mind a particular error E you do not want to exceed. You can
determine the sample size n you'll need if you have prior knowledge of the standard deviation sx. How would you know
the sample standard deviation in advance of the study? One way is to do a small "pre-study" to obtain an estimate of the
standard deviation. These are often called "pilot studies."
If we have an estimate of the standard deviation, then we can estimate the sample size needed to obtain the desired error
E.
Since E = tc*sx/√n, then solving for n yields = (tc*sx/E)²
Note that this is not a proper mathematical solution because t c is also dependent on n. While many texts use z c from the
normal distribution in the formula, we have not learned to calculate zc.
In the "real world" what often happens is that a result is found to not be statistically significant as the result of an initial
study. Statistical significance will be covered in more detail later. The researchers may have gotten "close" to statistical
significance and wish to shrink the confidence interval by increasing the sample size. A larger sample size means a
smaller standard error (n is in the denominator!) and this in turn yields a smaller margin of error E. The question is how
big a sample would be needed to get a particular margin of error E.
The value for tc from pilot study can be used to estimate the new sample size n. The resulting sample size n will be
slightly overestimated versus the traditional calculation made with the normal distribution. This overestimate, while
slightly unorthodox, provides some assurance that the error E will indeed shrink as much as needed.
In a study of body fat for 51 males students a sample mean x of 19.9 with a standard deviation of 7.7 was measured.
This led to a margin of error E of 2.17 and a confidence interval 17.73 ≤ μ ≤ 22.07
Suppose we want a margin of error E = 1.0 at a confidence level of 0.95 in this study of male student body fat. We can
use the sx from the sample of 51 students to estimate my necessary sample size:
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n = (2.0086*7.7/1)2 = 239.19 or 239 students. Thus I estimate that I will need 239 male students to reduce my margin of
error E to ±1 in my body fat study.
Other texts which use zc would obtain the result of 227.77 or 228 students. The eleven additional students would
provide assurance that the margin of error E does fall to 1.0.
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Hypothesis Testing
10.1 Confidence Interval Testing
In this chapter we explore whether a sample has a sample mean x that could have come from a population with a known
population mean μ. There are two possibilities. In Case I below, the sample mean x comes from the population with a
known mean μ. In Case II, on the right, the sample mean x does not come from the population with a known mean μ.
For our purposes the population mean μ could be a pre-existing mean, an expected mean, or a mean against which we
intend to run the hypothesis test. In the next chapter we will consider how to handle comparing two samples to each
other to see if they come from the same population.
Suppose we want to do a study of whether the female students at the national campus gain body fat with age during
their years at COM-FSM. Suppose we already know that the population mean body fat percentage for the new freshmen
females 18 and 19 years old is μ = 25.4.
We measure a sample size n = 12 female students at the national campus who are 21 years old and older and determine
that their sample mean body fat percentage is x = 30.5 percent with a sample standard deviation of sx = 8.7.
Can we conclude that the female students at the national campus gain body fat as they age during their years at the
College?
Not necessarily. Samples taken from a population with a population mean of μ = 25.4 will not necessarily have a sample
mean of 25.4. If we take many different samples from the population, the sample means will distribute normally about
the population mean, but each individual mean is likely to be different than the population mean.
In other words, we have to consider what the likelihood of drawing a sample that is 30.5 - 25.4 = 5.1 units away from
the population mean for a sample size of 12. If we knew more about the population distribution we would be able to
determine the likelihood of a 12 element sample being drawn from the population with a sample mean 5.1 units away
from the actual population mean..
In this case we know more about our sample and the distribution of the sample mean. The distribution of the sample
mean follows the student's t-distribution. So we shift from centering the distribution on the population mean and center
the distribution on the sample mean. Then we determine whether the confidence interval includes the population mean
or not. We construct a confidence interval for the range of the population mean for the sample.
If this confidence interval includes the known population mean for the 18 to 19 years olds, then we cannot rule out the
possibility that our 12 student sample is from that same population. In this instance we cannot conclude that the women
gain body fat.
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If the confidence interval does NOT include the known population mean for the 18 to 19 year old students then we can
say that the older students come from a different population: a population with a higher population mean body fat. In
this instance we can conclude that the older women have a different and probably higher body fat level.
One of the decisions we obviously have to make is the level of confidence we will use in the problem. Here we enter a
contentious area. The level of confidence we choose, our level of bravery or temerity, will determine whether or not we
conclude that the older females have a different body fat content. For a detailed if somewhat advanced discussion of this
issue see The Fallacy of the Null-Hypothesis Significance Test by William Rozeboom.
In education and the social sciences there is a tradition of using a 95% confidence interval. In some fields three different
confidence intervals are reported, typically a 90%, 95%, and 99% confidence interval. Why not use a 100% confidence
interval? The normal and t-distributions are asymptotic to the x-axis. A 100% confidence interval would run to plus and
minus infinity. We can never be 100% confident.
In the above example a 95% confidence interval would be calculated in the following way:
n = 12
x = 30.53
sx = 8.67
c = 0.95
degrees of freedom = 12 -1 = 11
tc = tinv((1-0.95,11) = 2.20
E = tc*sx/sqrt(12) = 5.51
x-E<μ<
+E
25.02 < μ < 36.04
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The 95% confidence interval for our n = 12 sample includes the population mean 25.3. We CANNOT conclude at the
95% confidence level that this sample DID NOT come from a population with a population mean μ of 25.3.
Another way of thinking of this is to say that 30.5 is not sufficiently separated from 25.8 for the difference to be
statistically significant at a confidence level of 95% in the above example.
In common language, the women are not gaining body fat.
The above process is reduced to a formulaic structure in hypothesis testing. Hypothesis testing is the process of
determining whether a confidence interval includes a previously known population mean value. If the population mean
value is included, then we do not have a statistically significant result. If the mean is not encompassed by the
confidence interval, then we have a statistically significant result to report.
Homework
If I expand my study of female students 21+ to n=24 and find a sample mean =28.7 and an sx=7, is the new sample
mean statistically significantly different from a population mean μ of 25.4 at a confidence level of c = 0.90?
10.2 Hypothesis Testing
The null hypothesis H0
The null hypothesis is the supposition that there is no change in a value from some pre-existing, historical, or expected
value. The null hypothesis literally supposes that the change is null, non-existent, that there is no change.
In the previous example the null hypothesis would have been H0: μ = 25.4
The alternate hypothesis H1
The alternate hypothesis is the supposition that there is a change in the value from some pre-existing, historical, or
expected value. Note that the alternate hypothesis does NOT say the "new" value is the correct value, just that whatever
the mean μ might be, it is not that given by the null hypothesis.
H1: μ ≠ 25.4
Statistical hypothesis testing
We run hypothesis test to determine if new data confirms or rejects the null hypothesis.
If the new data falls within the confidence interval, then the new data does not contradict the null hypothesis. In this
instance we say that "we fail to reject the null hypothesis." Note that we do not actually affirm the null hypothesis. This
is really little more than semantic shenanigans that statisticians use to protect their derriers. Although we run around
saying we failed to reject the null hypothesis, in practice it means we left the null hypothesis standing: we de facto
accepted the null hypothesis.
If the new data falls outside the confidence interval, then the new data would cause us to reject the null hypothesis. In
this instance we say "we reject the null hypothesis." Note that we never say that we accept the alternate hypothesis.
Accepting the alternate hypothesis would be asserting that the population mean is the sample mean value. The test does
not prove this, it only shows that the sample could not have the population mean given in the null hypothesis.
For two-tailed tests, the results are identical to a confidence interval test. Note that confidence interval never asserts the
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exact population mean, only the range of possible means. Hypothesis testing theory is built on confidence interval
theory. The confidence interval does not prove a particular value for the population mean , so neither can hypothesis
testing.
In our example above we failed to reject the null hypothesis H 0 that the population mean for the older students was
25.4, the same population mean as the younger students.
In the example above a 95% confidence interval was used. At this point in your statistical development and this course
you can think of this as a 5% chance we have reached the wrong conclusion.
Imagine that the 18 to 19 year old students had a body fat percentage of 24 in the previous example. We would have
rejected the null hypothesis and said that the older students have a different and probably larger body fat percentage.
There is, however, a small probability (less than 5%) that a 12 element sample with a mean of 30.5 and a standard
deviation of 8.7 could come from a population with a population mean of 24. This risk of rejecting the null hypothesis
when we should not reject it is called alpha α. Alpha is 1-confidence level, or α = 1-c. In hypothesis testing we use α
instead of the confidence level c.
Suppose
And we accept H0 as true
Reject H0 as false
H0 is true Correct decision. Probability: 1-α Type I error. Probability: α
H0 is false Type II error. Probability: β
Correct decision. Probability: 1 − β
Hypothesis testing seeks to control alpha α. We cannot determine β (beta) with the statistical tools you learn in this
course.
Alpha α is called the level of significance. 1-β is called the "power" of the test.
The regions beyond the confidence interval are called the "tails" or critical regions of the test. In the above example
there are two tails each with an area of 0.025. Alpha α = 0.05
For hypothesis testing it is simply safest to always use the t-distribution. In the example further below we will run a
two-tail test.
Steps
1.
2.
3.
4.
Write down H0, the null hypothesis
Write down H1, the alternate hypothesis
If not given, decide on a level of risk of rejecting a true null hypothesis H0 by choosing an α.
Determine the t-critical values from TINV(α,df).
5. Determine the t-statistic from
6. Make a sketch
7. If the t-statistic is "beyond" the t-critical values then reject the null hypothesis. By "beyond" we mean larger in
absolute value. Otherwise we fail to reject the null hypothesis.
Example
Using the data from the first section of these notes:
1. H0: μ = 25.4
2. H1: μ ≠ 25.4
- 84 -
3. Alpha α = 0.05 (α = 1-c, c = 0.95)
4. Determine the t-critical values: degrees of freedom: n-1=12-1; tc = TINV(α,df) = tinv(0.05,11) = 2.20
5. Determine the t-statistic
6. Make a sketch:
= (30.53-25.4)/(8.67/sqrt(8.67)) = 2.05
7. The t-statistic t is NOT "beyond" the t-critical values. We FAIL to reject the null hypothesis H0. We cannot say
the older female students came from a different population than the younger students with an population mean
of 25.4. Why not now accept H0: μ = 0 as the population mean for the 21 year old female students and older?
We risk making a Type II error: failing to reject a false null hypothesis. We are not trying to prove H 0 as being
correct, we are only in the business of trying to "knock it down."
Note the changes in the above sketch from the confidence interval work. Now the distribution is centered on μ with the
distribution curve described by a t-distribution with eleven degrees of freedom. In our confidence interval work we
centered our t-distribution on the sample mean. The result is, however, the same due to the symmetry of the problems
and the curve. If our distribution were not symmetric we could not perform this sleight of hand.
The hypothesis test process reduces decision making to the question, "Is the t-statistic t greater than the t-critical value
tc? If t > tc, then we reject the null hypothesis. If t < t c, then we fail to reject the null hypothesis. Note that t and t c are
irrational numbers and thus unlikely to ever be exactly equal.
Another example
I have a previously known population mean μ running pace of 6'09" (6.15). In 2001 I've been too busy to run regularly.
On my five most recent runs I've averaged a 6'23" (6.38) pace with a standard deviation 1'00" At an alpha α=0.05, am I
really running differently this year?
H0: μ = 6.15
H1: μ ≠ 6.15
Pay close attention to the above! We DO NOT write H1: μ = 6.23. This is a common beginning mistake.
1.
2.
3.
4.
H0: μ = 6.15
H1: μ ≠ 6.15
Alpha α = 0.05 (α = 1-c, c = 0.95)
Determine the t-critical values: degrees of freedom: n-1=5-1; tc = TINV(α,df) = tinv(0.05,4) = 2.78
5. Determine the t-statistic
6. Make a sketch:
= (6.38-6.15)/(1.00/sqrt(5)) = 0.51
- 85 -
7. The t-statistic t is NOT "beyond" the t-critical values. We FAIL to reject the null hypothesis H0.
Note that in my sketch I am centering my distribution on the population mean and looking at the distribution of sample
means for sample sizes of 5 based on that population mean. Then I look at where my actual sample mean falls with
respect to that distribution.
Note that my t-statistic t does not fall "beyond" the critical values. I do not have enough separation from my population
mean: I cannot reject H0. So I fail to reject H0. I am not performing differently than last year. The implication is that I
am not slower.
10.3 P-value
Return to our first example in these notes where the body fat percentage of 12 female students 21 years old and older
was =30.53 with a standard deviation sx=8.67 was tested against a null hypothesis H 0 that the population mean body
fat for 18 to 19 year old students was μ=25.4. We failed to reject the null hypothesis at an alpha of 0.05. What if we are
willing to take a larger risk? What if we are willing to risk a type I error rate of 10%? This would be an alpha of 0.10.
1.
2.
3.
4.
H0: μ = 25.4
H1: μ 25.4
Alpha α = 0.10 (α = 1-c, c = 0.90)
Determine the t-critical values: degrees of freedom: n-1=12-1; tc = TINV(α,df) = tinv(0.10,11) = 1.796
5. Determine the t-statistic
6. Make a sketch:
= (30.53-25.4)/(8.67/sqrt(8.67)) = 2.05
7. The t-statistic is "beyond" the t-critical value. We reject the null hypothesis H0. We can say the older female
students came from a different population than the younger students with an population mean of 25.4. Why not
now accept an H1: μ = 30.53 as the population mean for the 21 year old female students and older? We do not
actually know the population mean for the 21+ year old female students unless we measure ALL of the 21+
year old students.
- 86 -
With an alpha of 0.10 (a confidence interval of 0.90) our results are statistically significant. These same results were
NOT statistically significant at an alpha α of 0.05. So which is correct:
•
•
We FAIL to reject H0 because the t-statistic based on =30.53, μ=25.4, sx=8.76, is NOT beyond the critical
value for alpha α=0.05 OR
We reject H0 because the t-statistic based on =30.53, μ=25.4, sx=8.76, is beyond the critical value for alpha
α=0.10.
Note how we would have said this in confidence interval language:
•
We FAIL to reject H0 because μ=25.4 is within the 95% confidence interval for =30.53, sx=8.76 OR
•
We reject H0 because μ=25.4 is NOT within the 90% confidence interval for =30.53, sx=8.76.
The answer is that it depends on how much risk you are willing take, a 5% chance of committing a Type I error
(rejecting a null hypothesis that is true) or a larger 10% chance of committing a Type I error. The result depends on your
own personal level of aversity to risk. That's a heck of a mathematical mess: the answer depends on your personal
willingness to take a particular risk.
Consider what happens if someone decides they only want to be wrong 1 in 15 times: that corresponds to an alpha of α
= 0.067. They cannot use either of the above examples to decide whether to reject the null hypothesis. We need a system
to indicate the boundary at which alpha changes from failure to reject the null hypothesis to rejection of the null
hypothesis.
Consider what it would mean if t-critical were equal to the t-statistic. The alpha at which t-critical equals the t-statistic
would be that boundary value for alpha α. We will call that boundary value the p-value.
The p-value is the alpha for which tinv(α;df)=
. But how to solve for α?
The solution is to calculate the area in the tails under the t-distribution using the tdist function. The p-value is calculated
using the OpenOffice Calc formula:
=TDIST(ABS(t);degrees of freedom;number of tails)
The degrees of freedom are n − 1 for comparison of a sample mean to a known or pre-existing population mean μ.
Note that TDIST can only handle positive values for the t-statistic, hence the absolute value function.
p-value = TDIST(ABS(2.05;11;2) = 0.06501
The p-value represents the SMALLEST alpha α for which the test is statistically significant.
The p-value is the SMALLEST alpha α for which we reject the null hypothesis.
Thus for all alpha greater than 0.065 we reject the null hypothesis. The "one in fifteen" person would reject the null
hypothesis (0.0667 > 0.065). The alpha = 0.05 person would not reject the null hypothesis.
If the pre-chosen alpha is more than the p-value, then we reject the null hypothesis. If the pre-chosen alpha is less than
the p-value, then we fail to reject the null hypothesis.
The p-value lets each person decide on their own level of risk and removes the arbitrariness of personal risk choices.
Because many studies in education and the social sciences are done at an alpha of 0.05, a p-value at or below 0.05 is
used to reject the null hypothesis.
- 87 -
1 − p-value is the confidence interval for which the new value does not include the pre-existing population mean.
Another way to say this is that 1 − p-value is the maximum confidence level c we can have that the difference (change)
is significant. We usually look for a maximum confidence level c of 0.95 (95%) or higher.
Note:
p-value = TDIST(ABS(t),n−1,number_tails)
10.4 One Tailed Tests
All of the work above in confidence intervals and hypothesis testing has been with two-tailed confidence intervals and
two-tailed hypothesis tests. There are statisticians who feel one should never leave the realm of two-tailed intervals and
tests.
Unfortunately, the practice by scientists, business, educators and many of the fields in social science, is to use one-tailed
tests when one is fairly certain that the sample has changed in a particular direction. The effect of moving to a one tailed
test is to increase one's risk of committing a Type I error.
One tailed tests, however, are popular with researchers because they increase the probability of rejecting the null
hypothesis (which is what most researchers are hoping to do).
The complication is that starting with a one-tailed test presumes a change, as in ANY change in ANY direction has
occurred. The proper way to use a one-tailed test is to first do a two-tailed test for change in any direction. If change has
occurred, then one can do a one-tailed test in the direction of the change.
Returning to the earlier example of whether I am running slower, suppose I decide I want to test to see if I am not just
performing differently (≠), but am actually slower (<). I can do a one tail test at the 95% confidence level. Here alpha
will again be 0.05. In order to put all of the area into one tail I will have to use the OpenOffice Calc function
TINV(2*α,df).
H0: μ = 6.15
H1: μ < 6.15
μ=6.15
x = 6.38
sx = 1.00
n=5
degrees of freedom (df)=4
tc = TINV(2*α,df) = TINV(2*0.05,4) = 2.13
t-statistic =
= (6.38-6.15)/(1.00/sqrt(5)) = 0.51
Note that the t-statistic calculation is unaffected by this change in the problem.
Note that my t-statistic would have to exceed only 2.13 instead of 2.78 in order to achieve statistical significance. Still,
0.51 is not beyond 2.13 so I still DO NOT reject the null hypothesis. I am not really slower, not based on this data.
- 88 -
Thus one tailed tests are identical to two-tailed tests except the formula for tc is TINV(2*α,df) and the formula for p is
=TDIST(ABS(t),n−1,1).
Suppose we decide that the 30.53 body fat percentage for females 21+ at the college definitely represents an increase.
We could opt to run a one tailed test at an alpha of 0.05.
1.
2.
3.
4.
H0: μ = 25.4
H1: μ ≠ 25.4
Alpha α = 0.05 (α = 1-c, c = 0.95)
Determine the t-critical values: degrees of freedom: n-1=12-1; tc = TINV(2*α,df) = tinv(2*0.05,11) = 1.796
5. Determine the t-statistic
6. Make a sketch:
= (30.53-25.4)/(8.67/sqrt(8.67)) = 2.05
7. The t-statistic is "beyond" the t-critical value. We reject the null hypothesis H0. We can say the older female
students came from a different population than the younger students with an population mean of 25.4. Why not
now accept an H1: μ = 30.53 as the population mean for the 21 year old female students and older? We do not
actually know the population mean for the 21+ year old female students unless we measure ALL of the 21+
year old students.
8. The p value is =TDIST(ABS(2.05),11,1)=0.033
This result should look familiar: it is the result of the two tail test at alpha = 0.10, only now we are claiming we have
halved the Type I error rate (α) to 0.05. Some statisticians object to this saying we are attempting to artificially reduce
our Type I error rate by pre-deciding the direction of the change. Either that or we are making a post-hoc decision based
on the experimental results. Either way we are allowing assumptions into an otherwise mathematical process. Allowing
personal decisions into the process, including those involving α, always involve some controversy in the field of
statistics.
=TDIST(0.51,4,2) = 0.64
10.5 Hypothesis test for a proportion
For a sample proportion p and a known or pre-existing population proportion P, a hypothesis can be done to determine if
the sample with a sample proportion p could have come from a population with a proportion P. Note that in this text,
due to typesetting issues, a lower-case p is used for the sample proportion while an upper case P is used for the
population proportion.
In another departure from other texts, this text uses the student's t-distribution for tc providing a more conservative
determination of whether a change is significant in smaller samples sizes. Rather than label the test statistic as a zstatistic, to avoid confusing the students and to conform to usage in earlier sections the test statistic is referred to as a tstatistic.
A survey of college students found 18 of 32 had sexual intercourse. An April 2007 study of abstinence education
programs in the United States reported that 51% of the youth, primarily students, surveyed had sexual intercourse. Is the
proportion of sexually active students in the college different from that reported in the abstinence education program
study at a confidence level of 95%?
The null and alternate hypotheses are written using the population proportion, in this case the value reported in the
study.
- 89 -
H0: P = 0.51
H1: P ≠ 0.51
sample proportion p = 18/32 = 0.5625
sample proportion q = 1 − p = 0.4375
Note that n*p must be > 5 and n*q must also be > 5 just as was the case in constructing a confidence interval.
Confidence level c = 0.95
The t-critical value is still calculated using alpha α along with the degrees of freedom:
=TINV(0.05;32−1)
=2.04
The only "new" calculation is the t-statistic t:
Note that the form is still (sample statistic - population parameter)/standard error for the statistic.
=(0.5625-0.51)/SQRT(0.5625*0.4375/32)
=0.5545
The t-statistic t does not exceed the t-critical value, so the difference is not statistically significant. We fail to reject the
null hypothesis of no change.
The p-value is calculated as above using the absolute value of the t-statistic.
=TDIST(ABS(0.5545);32-1;2)
=0.58
The maximum level of confidence c we can have that this difference is significant is only 42%, far too low to say there
is a difference.
The formulas below are for OpenOffice.org Calc. If using Excel, change the semi-colons to commas.
Statistic or Parameter
Square root
Sample size
Sample mean
Symbol
Equations
Basic Statistics
√
n
x
σx/n
OOo
=SQRT(number)
=COUNT(data)
=AVERAGE(data)
Sample standard deviation
sx
Sample Coefficient of Variation
CV
(sx/x)
=STDEV(data)/AVERAGE(
data)
Calculate a z-score from an x
z
=
=STANDARDIZE(x,μ,σ)
Slope
=STDEV(data)
Linear Regression
b
Intercept
a
Correlation
r
Coefficient of Determination
r2
Normal Statistics
Calculate an area p to the left of x from an x value.
p
Calculate an x from an area p to the left of x
x
Confidence interval statistics
Degrees of freedom
df
= n-1
- 90 -
=SLOPE(y data;x data)
=INTERCEPT(y data;x
data)
=CORREL(y data;x data)
=(CORREL(y data;x
data))^2
=NORMDIST(x;μ;σ;1)
=NORMINV(p,μ;σ)
Find the limit for a confidence interval for n ≥ 30 using a normal distribution with p as the
area to the left of the limit
Find the lower limit for a confidence interval for n ≥ 30 using a normal distribution for a
confidence level c
Find the upper limit for a confidence interval for n ≥ 30 using a normal distribution for a
confidence level c
Degrees of freedom
df
=n−1
Find a tc (t-critical) value from a confidence level c and
tc
sample size n
Standard error of the mean
Calculate the margin of error E for a mean for any n ≥ 5
E
Error tolerance
using sx.
Calculate a confidence interval for a population mean μ from the
x−E≤μ≤x+E
sample mean x and margin of error E for the mean.
Hypothesis Testing
Relationship between confidence level c and alpha α for
1−c=α
two-tailed tests
=NORMINV(p;μ;σ/SQRT(n
))
=NORMINV((1c)/2;μ;σ/SQRT(n))
=NORMINV((1+c)/2;μ;σ/S
QRT(n))
=COUNT(data)-1
Calculate a t-statistic (t)
=(x − μ)/(sx/SQRT(n))
t
Calculate t-critical for a two-tailed test for a risk of a type t
c
I error alpha α
tc
Calculate t-critical for a one-tailed test
Calculate a p-value from a t-statistic t
p
- 91 -
=TINV(1-c;n-1)
=sx/SQRT(n)
=tc*sx/SQRT(n)
=TINV(α,df)
=TINV(2*α,df)
= TDIST(ABS(t);df;#tails)
Testing the two sample means with the t-test
11.1 Paired differences: Dependent samples
Many studies investigate systems where there are measurements taken before and after. Usually there is an experimental
treatment or process between the two measurements. A typical such system would be a pre-test and a post-test.
Inbetween the pre-test and the post-test would typically be an educational or training event. One could examine each
student's score on the pre-test and the post-test. Even if everyone did better on the post-test, one would have to prove
that the difference was statistically significant and not just a random event.
These studies are called "paired t-tests" or "inferences from matched pairs". Each element in the sample is considered as
a pair of scores. The null hypothesis would be that the average difference for all the pairs is zero: there is no difference.
For a confidence interval test, the confidence interval for the mean differences would include zero if there is no
statistically significant difference.
If the difference for each data pair is referred to as d, then the mean difference could be written d. The hypothesis test is
whether this mean difference d could come from a population with a mean difference μ d equal to zero (the null
hypothesis). If the mean difference d could not come from a population with a mean difference μd equal to zero, then
the change is statistically significant. In the diagram above the mean difference μd is equal to μbefore − μ after.
Confidence interval test
Consider the paired data below. The first column are female body fat measurements from the beginning of a term. The
second column are the body fat measurements sixteen weeks later. The third column is the difference d for each pair.
BodyFat before
23.5
28.9
29.2
24.7
26.4
Bodyfat after
20.8
27.5
28.4
24.1
26.1
Bodyfat difference d
-2.7
-1.4
-0.8
-0.6
-0.3
- 92 -
23.7
24
0.3
46.9
47.2
0.3
23.6
24
0.4
26.4
27.1
0.7
15.9
17
1.1
30.3
31.5
1.2
28.0
29.3
1.3
36.2
37.6
1.4
31.3
32.8
1.5
31.5
33.2
1.7
26.7
28.6
1.9
26.5
29.0
2.5
The confidence interval is calculated on the differences d (third column above) using the sample size n, sample mean
difference d, and the sample standard deviation of the difference data d. The following table includes calculations using
a 95% confidence interval.
Count of the differences
sample mean difference d
Standard deviation of the difference data d
Standard error for the mean of the difference data d
tc for confidence level = 0.95
17
0.50
1.33
0.32
2.12
Margin of the error E for the mean
0.68
Lower bound for the 95% confidence interval
-0.18
Upper bound for the 95% confidence interval
1.18
The 95% confidence interval includes a possible population mean of zero. The population mean difference μd could be
equal to zero.
This means that "no change" is a possible population mean. To use the double negative, we cannot rule out the
possibility of no change. We fail to reject the null hypothesis of no change. The women have not statistically
significantly gained body fat over the sixteen weeks of the term.
Hypothesis test
Spreadsheets provide a function to calculate the p-value for paired data using the student's t-distribution. This function
is the TTEST function. If the the p-value is less than your chosen risk of a type I error α then the difference is
significant. The function does not require generating the difference column d as seen above, only the original data is
used in this function.
The function takes as inputs the before data (data_range_x), the after data (data_range_y), the number of tails, and a
final variable that specifies the type of test. A paired t-test is test type number one.
=TTEST(data_range_x;data_range_y;2;1)
To ensure that the spreadsheet calculates the p-value correctly, delete any data missing the x or y value in the pair. Data
missing an x or y value is not paired data!
The maximum confidence we can have that the difference is statistically significant is 1 − p-value.
p-value
0.14
Maximum confidence level c 0.86
The p-value confirms the confidence interval analysis, we fail to reject the null hypothesis. At a 5% risk of a type I error
we would fail to reject the null hypothesis. We can have a maximum confidence of only 86%, not the 95% standard
typically employed. Some would argue that our concern for limited the risk of rejecting a true null hypothesis (a type I
error) has led to a higher risk of failing to reject a false null hypothesis (a type II error). Some would argue that because
of other known factors - the high rates of diabetes, high blood pressure, heart disease, and other non-communicable
diseases - one should accept a higher risk of a type I error. The average shows an increase in body fat. Given the short
- 93 -
time frame (a single term), some might argue for reacting to this number and intervening to reduce body fat. They
would argue that given other information about this population's propensity towards obesity, 86% is "good enough" to
show a developing problem. Ultimately these debates cannot be resolved by statisticians.
11.2 T-test for means for independent samples
One of the more common situations is when one is seeking to compare two independent samples to determine if the
means for each sample are statistically significantly different. In this case the samples may differ in sample size n,
sample mean, and sample standard deviation.
In this text the two samples are refered to as the x data and the y data. The sample size for the x data is n x. The sample
mean for the x data is x. The sample standard deviation for the x data is s x. For the y data, the sample size is n y, the
sample mean is y, and the sample standard deviation is sy.
Two possibilities exist. Either the two samples come from the same population and the population mean difference is
statistically zero. Or the two samples come from different populations where the population mean difference is
statistically not zero.
Confidence Interval test
Each sample has a range of probable values for their population mean μ. If the confidence interval for the sample mean
differences includes zero, then there is no statistically significant difference in the means between the two samples. If
the confidence interval does not include zero, then the difference in the means is statistically significant.
Note that the margin of error E for the mean difference is still t c multiplied by the standard error. The standard error
formula changes to account for the differences in sample size and standard deviation.
Thus the margin of error E can be calculated using:
For the degrees of freedom in the t-critical t c calculation use n − 1 for the sample with the smaller size. This produces a
- 94 -
conservative estimate of the degrees of freedom. Advanced statistical software uses another more complex formula to
determine the degrees of freedom.
The confidence interval is calculated from:
(x − y) − E < (μx − μy) < (x − y) + E
Where x is the sample mean of one data set and y is the sample mean of the other data. Some texts use the symbol xd
for this difference and μd for the hypothesized difference in the population means. This leads to the more familiar
looking formulation:
xd − E < μd < xd + E
Where:
μd = μx − μy and
xd = x − y
Remember, μx and μy are not known. These are left as symbols. After calculating the interval, check to see if the
confidence includes zero. If zero is inside the interval, then the sample means are not significantly different and we fail
to reject the null hypothesis.
The following table uses a local business example. Data was recorded as to how many cup of sakau were consumed in a
single night at two sakau markets on Pohnpei. The implication is that the lower the average, the stronger the sakau. The
average difference, however, has to be statistically significant.
Song mahs (x)
2
3
6
3
3.5
4.5
1
5
3
7
4
2.5
5.5
2
1
2
Rush Hour (y)
2
10
1.5
5.5
9
7.5
5.5
3
3
6
3
4.5
10
9
2
2
4
5
5
5.5
15
14
2
2
4
Sample statistics
sample size n 16
25
sample mean 3.44
5.6
sample stdev 1.77
3.73
Confidence interval statistics
standard error
0.87
t-critical tc
2.13
- 95 -
margin of error E
difference of means
lower bound ci
upper bound ci
Note that 15 was used for the
sample.
1.85
-2.16
-4.01
-0.31
degrees of freedom in the t-critical calculation. Sixteen is the sample size of the smaller
Note that the confidence interval does not include zero. The confidence interval indicates that whatever the population
mean difference μd might be, the population mean μd cannot be zero. This means that the sample means are statistically
significantly different. We would reject a null hypothesis of no difference between the two markets. The implication is
that Song Mahs is stronger than Rush Hour, at least on these two nights.
Hypothesis test
As noted above, spreadsheets provide a function to calculate p-values. If the the p-value is less than your chosen risk of
a type I error α then the difference is significant.
The function takes as inputs one the data for one if the two samples (data_range_x), the data for the other sample
(data_range_y), the number of tails, and a final variable that specifies the type of test. A t-test for means from
independent samples is test type number three.
=TTEST(data_range_1;data_range_2;number of tails;3)
For the above data, the p-value is given in the following table:
p-value
0.02
Maximum confidence level c 0.98
The TTEST function does not use the smaller sample size to determine the degrees of freedom. The TTEST function
uses a different formula that calculates a larger number of degrees of freedom, which has the effect of reducing the pvalue. Thus the confidence interval result could produce a failure to reject the null hypothesis while the TTEST could
produce a rejection of the null hypothesis. This only occurs when the p-value is close to your chosen α.
[Optional material!] If you have doubts and want to explore further, take the difference of the means and divide by the
standard error to obtain the t-statistic t. Then use the TDIST function to determine the p-value, using the smaller sample
size − 1 to calculate the degrees of freedom.
Note that (μx − μy) is presumed to be equal to zero. Thus the formula is the difference of the means divided by the
standard error (given further above).
t = xd ÷ (standard error)
Once t is calculated, use the TDIST function to determine the p-value.
=TDIST(ABS(t);n−1;2)
Technical side-note: TTEST type three does not presume that the population standard deviations σ x and σy are equal.
This is in keeping with modern practice and reality. TTEST type two presumes σx = σy. One rarely knows either value,
and if one did know those values, why would not they also know the actual population means? With the true population
means in hand, then any difference would be significant.
- 96 -